There are some good answers about Southern views of slavery and racial hierarchy by u/Georgy_K_Zhukov, and the FAQ has a few more slavery-related answers.
But your question goes to a very common trope that pops up when it comes to the Confederacy and those who fought for it. Namely, that eventually, they'd either see the error of their ways, or potentially just lose interest in the institution of slavery. There's a whole subsection of Confederate apologia that tries to show how the "great men" of the Confederate Army, guys like Robert E Lee, Stonewall Jackson, and others, really actually didn't like slavery and thought that they could somehow deal with it later, but had to fight for their state/country first. Of course, like most Confederate apologia, this ignores the elephant in the room: all those black people.
There were millions of black people, most of whom were slaves, in the Confederate South. Had there been a plan to phase out slavery, or even an inkling that this might happen eventually, there would have been talk about what exactly was going to happen with all of those former slaves. Abolitionists had thought about this extensively, with many (including Lincoln) pondering the idea of either sending them back to Africa or somewhere else. If you didn't do that, of course, you'd need to integrate them into your own society, recognize their rights, perhaps even allow them to vote and own property and, you know, be human.
Speaking of which, you'd also expect to see some recognition that slavery was wrong on the basis of basic human rights. In reality, we see the exact opposite from Confederate leaders and politicians. The single most cited quote for this is from the Vice President of the Confederacy, Alexander Stephens, who neatly summed up what his new country was all about and why he was a total POS when he said
Our new government is founded upon exactly the opposite idea; its foundations are laid, its corner-stone rests, upon the great truth that the negro is not equal to the white man; that slavery subordination to the superior race is his natural and normal condition. This, our new government, is the first, in the history of the world, based upon this great physical, philosophical, and moral truth.
What was that opposite idea that he's talking about? Well, pretty much exactly what you're asking about:
The prevailing ideas entertained by [Jefferson] and most of the leading statesmen at the time of the formation of the old constitution, were that the enslavement of the African was in violation of the laws of nature; that it was wrong in principle, socially, morally, and politically. It was an evil they knew not well how to deal with, but the general opinion of the men of that day was that, somehow or other in the order of Providence, the institution would be evanescent and pass away.
A few things are clear from this, the famous "Cornerstone Speech." One is that the VP of the new government was staunchly against the idea of phasing out slavery. In fact, this was the fear that launched the thousands ships of secession in the first place. What you're asking about is more or less what many abolitionists in the North were trying to do, and what leaders in the South were trying to stop.
More importantly, the prevailing view among those who defended slavery was that it was the natural subordination of an inferior race to a superior one. In their twisted way, they turned slavery into a moral good. Once you see it this way, there's no room to phase out the institution. In their eyes, black people were not capable of being free, and could not actually function as equal members of society. Regardless of the obviously ample economic interest that the South had in keeping slavery going, this white supremacist ideology meant that even those who did not own slaves themselves saw slavery as the proper way of the world.
This is the biggest problem with any argument that claims that the South would have eventually left slavery behind. Again, the economic benefits of free labor alone may have been enough to sustain the institution of slavery. But beyond that, by and large, people in the South (and for that matter, many in the North) did not see slaves as people worthy of equality, freedom, or even recognition as actual people. With this mindset, what do you do with all these former slaves, who you "know" are inferior and cannot contribute as free men and women to the great society that you (white people) have built? If anything, you figure that you're doing them a "favor" by keeping them in servitude, teaching them the ways of Christianity, and trying to reform their "primitive" ways as best you can.
It's worth reading through some relevant sections of various states' declarations of secession. One thing that becomes immediately clear when you read them is that, well, how about Mississippi explains it:
Our position is thoroughly identified with the institution of slavery-- the greatest material interest of the world. Its labor supplies the product which constitutes by far the largest and most important portions of commerce of the earth. These products are peculiar to the climate verging on the tropical regions, and by an imperious law of nature, none but the black race can bear exposure to the tropical sun. These products have become necessities of the world, and a blow at slavery is a blow at commerce and civilization. That blow has been long aimed at the institution, and was at the point of reaching its consummation. There was no choice left us but submission to the mandates of abolition, or a dissolution of the Union, whose principles had been subverted to work out our ruin. That we do not overstate the dangers to our institution, a reference to a few facts will sufficiently prove.
Georgia:
The prohibition of slavery in the Territories, hostility to it everywhere, the equality of the black and white races, disregard of all constitutional guarantees in its favor, were boldly proclaimed by its leaders and applauded by its followers.
With these principles on their banners and these utterances on their lips the majority of the people of the North demand that we shall receive them as our rulers.
The prohibition of slavery in the Territories is the cardinal principle of this organization.
Texas:
In all the non-slave-holding States, in violation of that good faith and comity which should exist between entirely distinct nations, the people have formed themselves into a great sectional party, now strong enough in numbers to control the affairs of each of those States, based upon an unnatural feeling of hostility to these Southern States and their beneficent and patriarchal system of African slavery, proclaiming the debasing doctrine of equality of all men, irrespective of race or color-- a doctrine at war with nature, in opposition to the experience of mankind, and in violation of the plainest revelations of Divine Law. They demand the abolition of negro slavery throughout the confederacy, the recognition of political equality between the white and negro races, and avow their determination to press on their crusade against us, so long as a negro slave remains in these States.
All emphases mine. You can read them in full along with a few other states here.
As you can see, slavery was not merely a facet of Southern culture or an institution within the Confederacy. It was its entire raison d'être, and white supremacist ideology was its core. Phasing out slavery was exactly what the seceding states feared was happening in the US, and they seceded specifically in order to stop that phasing out. Furthermore, they viewed slavery as an inherently good system, one which kept black and white races in their respective places in society. There was no sense that black people had their own rights, and no plan or thought as to how those rights might be recognized if slavery were to be phased out.
/u/DGBD mentioned a few answers I've written in the past, and touched on a few issues themself, but I'll also repost this older response I wrote up which focuses specifically on the issue of Emancipation and the Confederacy.
So there are essentially two things that are in play here. It is true that Historians debate whether or not slavery would have come to an end in the South had the Confederacy won, if the Civil War had been avoided, or if hostilities had been ended swiftly, prior to the Emancipation Proclamation with some sort of agreement that guaranteed its continued existence. As with any counterfactual discussion, there is no guaranteed answer, and you can find real, scholarly debate about just how long the 'shelf-life' of slavery was. (See, for instance "Would Slavery Have Survived Without the Civil War? Economic Factors in the American South During the Antebellum and Postbellum Eras" in Southern Cultures Summer 2013, which is a 'for'-'against' discussion between Stanley L. Engerman and Peter A. Coclanis).
But that isn't really the question at hand. The question here is whether there were clear, defined plans within the Confederacy to end slavery while 'winning', and such an assertion is outright laughable. At its founding, the Confederacy made no secret of how important slavery was to it. Pres. Davis in an early address to Congress, noted "the labor of African slaves was and is indispensable" to the success of the Confederacy. To be sure, there were a few proposals, but they bear no resemblance to this plan related above (which, ironically, sounds much more akin to the Emancipation plans advanced in the North). I've written previously about the centrality of slavery to the Confederacy's reason for existence, which, aside from, I hope, offering a fair window into just why the idea that the South would immediately be contemplating getting rid of slavery within only a year or two of leaving the Union to protect the institution, additionally provides some specifics for just why the above statement is so off-kilter. For starters, such a law would be Unconstitutional, since, while nearly a wholesale copy of the American Constitution, most of the changes, most especially Article I, Sec. 9(4) - "No bill of attainder, ex post facto law, or law denying or impairing the right of property in negro slaves shall be passed" - are focused on the protection of their 'peculiar institution'. The most advanced plan which can be called "Emancipation" would likely be the Barksdale Bill, which was passed in March of 1865, and did allow for the enlistment of black slaves, hanging the carrot of freedom in front of them, but it was an abject failure. Not only did barely a handful of willing men show up, as they knew there was nothing in it for them as the war was nearly lost anyways, but it also helps illustrate just how divisive the proposal remained, if the retorts of those who opposed the bill are any indication. Earlier in the war, when a similar proposal had been made by Gen. Patrick Cleburne, it had essentially killed his career, given how unpopular the idea was.
And to be clear, the Emancipation in both of these plans was strictly as a reward for military service, not to extend beyond that. What other proposals that existed, which to reiterate, by no stretch of the definition can be called "well managed plan" let alone devised "by mid-war, when they were on the winning side of things", were embryonic. During the debates on what became the Barksdale Bill, for instance, avowed slavery advocate John Stringfellow, did propose a broad emancipation plan, but it was a desperate bid, in the face of impending defeat (Stringfellow writing to Davis in the winter of 1865), to preserve the system of plantation slavery in all but name, arguing that "if we emancipate, our independence is secured, the white man only will have any and all political rights” and whites' power over the blacks would thus remain "on terms about as economical as though owned by him.” It was not an idea unique to Stringfellow, at that point in time, as others had expressed similar sentiments in private correspondence. It is, of course, supremely ironic, since while such a bid, had it been enacted, would be unlikely to succeed, given the Union's position of strength at the negotiating table that late in the war, but of course, in the end, the vain hope of some Confederate leaders, expressed by Stringfellow, would nevertheless essentially triumph in the form of Jim Crow, which while not exact in the details, certainly encapsulated the sentiments.
So to sum this up, it is not wrong to say that there were proposals for Emancipation within the Confederacy, but it is absolutely wrong to say that there was "a well managed plan to eradicate the practice of slavery by mid-war, when they were on the winning side of things". There was no such plan in 1862-1863, and when even such a limited proposal as Cleburne's was made in 1863, it was shot down immediately. In Davis' words:
the dissemination or even promulgation of such opinions under the present circumstances of the Confederacy, whether in the Army or among the people can be productive only of discouragement, distraction, and dissension.
When such proposals, whether of partial or total Emancipation, did finally become more politically tenable, it was only in 1865, when the Confederacy was on the ropes and its leadership fully aware that they were making desperate, hail Mary bids for survival, hardly legislating while on 'the winning side of things'.
For further reading on the topic of Emancipation within the Confederacy, I would recommend "Confederate Emancipation; Southern Plans to Free and Arm Slaves during the Civil War" by Bruce Levine.
Matt Karp in This Vast Southern Empire: Slaveholders at the Helm of American Foreign Policy, a book about the antebellum foreign policy of the United States, discusses how there were several ambitious but completely unrealistic plans for either the Southern US or the US as a whole to create a political union of all the remaining slave-holding countries in the Americas (which at the time mainly meant Cuba and Brazil) and then expand to encompass a vast circle that would span the entire Caribbean Basin and Gulf of Mexico, from the US down to the northern half of South America. The US (or maybe just the southern half), plus all of Mexico and Central America, all of the Caribbean islands, plus Colombia, Venezuela, Brazil, and a few other South American nations would become part of a vast white supremacist empire, built on plantation slavery.
Even the most delusional slaveholder-diplomats didn't envision this happening any time soon, but there were serious efforts by the US before the war, and the Confederacy during the war, to annex Cuba, so we know some version of this plan was seriously contemplated.
As for the vision of slavery's future, Confederate ideologues were always vague, some would say intentionally. Some went with the benevolence argument that slavery was necessary to "civilize" Africans, and once they had been fully civilized (which some said might happen fairly soon, and some said might take many decades or even centuries), they could be freed and live like whites. Others, like John C. Calhoun, argued more forcefully that slavery was "not a necessary evil, but a positive good"[2] and that subordination of the African to the European was "his natural condition", possibly implying slavery should go on in perpetuity. He argued Southern slavery was in fact a better system than existed in the North, and said slaves lived better than many free laborers in the North, because slave owners were required to feed and clothe their slaves, and care for them in old age, whereas free laborers could treated as entirely disposable. In either case, both types of ideologue supported the expansion of slavery to new territories in the west, such as Texas and Kansas. None took the type of compromise position of Abraham Lincoln who argued slavery should be limited to the Southern states where it already existed and not allowed to expand, with the hope that if it could not grow, it might gradually decay as an institution.
It might be most accurate to answer your question by simply saying "they weren't thinking that far ahead." Some claimed to envision a day when slavery would be gone, while others did not explicitly say it would or should end one day. But neither of them envisioned it ending soon. Neither envisioned it being something they needed to start preparing for. If they thought it should happen, it would be far enough away that it was someone else's responsibility to figure it out, they weren't planning to do it themselves.
Sources:
Karp, Matt; "This Vast Southern Empire" Harvard University Press, 2016
Calhoun, John; "Slavery a Positive Good" United States Senate, 1837
In addition to the wonderful answers by /u/Georgy_K_Zhukov and /u/DGBD I'd like to mention the example of comparative emancipation in the Americas. No American nation reliant on slavery perpetuated the institution beyond the 1890s.
Brazil and Cuba (while still a Spanish colony) emancipated their slaves. Why? There was no one cause but several major forces. 1) rising political and economic liberalism painted slavery as an archaic institution and one incompatible with modern capitalism especially the need for a free market in labor (not for the benefit of workers but for the benefit of employers) 2) the persistent cost and difficultly of policing slavery and social control. This was especially true in Cuba as anti-slavery resistance became intertwined with nascent independence movements. Abolition was a means to undercut pressures for independence.
The biggest difference between the American south and those two slave countries was that the slave population in the US became self reproducing far earlier. This meant that before the end of the transatlantic trade an internal slave trade existed allowing for the expansion of slavery domestically, particularly from the deep South to the west. Brazil and Cuba only began to consider domestic reproduction after the end of the international trade, but that still came 60 years before eventual abolition.
The US South was not so unique that these factors would not have eventually come into play. Some constellation of those forces and other domestic issues ended slavery in every American nation. A Confederate States of America would have been no different. The timing may have been different but the outcome the same.