There were 5 (or if you consider post-new-era deal 6) party systems in the us-history. 3rd and 4th party systems were eras of republican dominance, from the 1860s to the 1920s. They were, like you mentioned, the progressive party, alligned to northern laborers, minorities, capital and immigrants, while the democrats sided themselves with the poor white farmers, mostly in the southern and midwest states.

It was F. D. Roosevelt who formed a party program during the great depression that made a wild coalition of diverse (even opposing) social groups possible, named "new deal coalition". This coalition of voters made the success of the democratic party even after fdr possible, while the gop lost ground and focused in 2 major shifts of programs onto liberals (in the economic understanding), christians and conservatives. This made the shift of "progressive parties" possible and these new values and programs forged the modern party programs.

Im sorry for my english, im a bit out of practise. I hope this was helpful, but if you need more information feel free to ask :)

The problem is that this is using "conservative" and "progressive" in relative terms instead of as ideological constants. By the definition of "conservative," conservatism seeks to generally maintain the current establishments. In contrast, progressivism seeks to move beyond the current state.

When we think of 1860s conservatism in the United States, that which was sought for preservation were largely states' rights and the institution of slavery. Perhaps the best document we can use to show what was being "conserved" is the Mississippi Declaration of Secession:

Our position is thoroughly identified with the institution of slavery - the greatest material interest of the world…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…Utter subjugation awaits us in the Union, if we should consent longer to remain in it. It is not a matter of choice, but of necessity. We must either submit to degradation…or we must secede from the Union framed by our fathers, to secure this as well as every other species of property."

So whereas the Republican Party specifically stated that it was in opposition to the expansion of slavery—

"…we deny the authority of Congress, of a territorial legislature, or of and individuals, to give legal existence to slavery in any territory of the United States."

—the Democratic Party controlled areas of the Deep South were fighting to maintain that institution.

The "switch" to which many people refer and about which you've asked is most commonly placed as occurring in the 1960s under the administration of Lyndon Johnson. The difficulty with this is calling it an actual switch in platform. There is little to no evidence that the Republican Party became the bastion of the former Jim-Crowe southerners. Their ideology was simply dying off as represented by an Old Guard of the Democratic Party represented by such people as George Wallace, Bull Connor, Strom Thurmond, and Robert Byrd.

When we use conservative and liberal in the contemporary sense, we're talking about a very different thing than we were in 1860. The Republican Party became "conservative" as they became a party focused on maintaining the institutions prior to FDR's New Deal. Since the era of Reagan especially, the Republicans have been characterized with an undying devotion to the free market. Since the time of FDR, the Democratic Party began to shift toward what he once called "enlightened" government, or the creation of a large, centralized welfare state.

The Republicans seem to have moved more heavily toward the idea of free markets since the nomination of Barry Goldwater in 1964. He became an influence on the later Reagan administration which would draw a contrast distinctly from the Eisenhower administration's maintaining of the massive tax system and the New Deal programs which Roosevelt had instituted. To show the thoughts of Eisenhower (which dramatically contrast with the perceived notion of the Republican Party through the 80s):

"Should any political party attempt to abolish social security, unemployment insurance and eliminate labor laws and farm programs, you would not hear of that party again in our political history. There is a tiny splinter group, of course, that believes that you can do these things…Their number is negligible and they are stupid."

The conservative–liberal switch largely comes down to a shift in focus of issues and it largely occurred between the time of FDR and Barry Goldwater. While the Democrats began to pursue the expansion of the state towards an ideal of welfare-state liberalism, the Republicans, especially under Goldwater and later Reagan, sought back to the days prior to the New Deal and took up the banner of the laissez-faire liberalism (or modern American conservatism) that they associated with past figures such as Calvin Coolidge.

Sources:

-Mississippi Declaration of Causes of Secession, January 9, 1861.

-Republican Party, Republican Party Platform of 1860, sections 7–10.

-Dwight Eisenhower, letter to Edgar Eisenhower, November 8, 1954.

-Franklin Delano Roosevelt, "Commonwealth Club Address," September 23, 1932, accessed September 14, 2016.

-Republican Party, Republican Party Platform of 1964, section 2, subsection on Fiscal Irresponsibility.

-John Hendrickson, "Calvin Coolidge and Ronald Reagan," Calvin Coolidge Presidential Foundation, October 20, 2014, accessed September 14, 2016.