My area of specialty is ancient Greece and Rome, so that is the area I will focus on for my reply here. Other users may be able to share information about possible abolitionists in other areas of the ancient world.

I wrote a post on my blog three years ago in October 2019 in which I try to answer the question of whether there were people who might be described as abolitionists in ancient Greece and Rome. The answer is, as far as I am aware, not really; there were definitely some people in ancient Greece and Rome who believed that slavery is morally wrong, but there is no evidence for the existence of any kind of sustained, organized campaign to abolish slavery at an institutional level in the way there was, say, in the United States in the nineteenth century.

As far as I am currently aware, there is only one surviving mention of the existence of people in Classical Greece who believed that slavery is categorically unjust. Namely, the Greek philosopher Aristotle of Stageira (lived 384 – 322 BCE) very briefly mentions in his Politics 1.1253b that there were people in his own time in the Greek world in the fourth century BCE who believed that slavery is "contrary to nature" and therefore "unjust." Aristotle writes, as translated by H. Rackham:

“For some thinkers hold the function of the master to be a definite science, and moreover think that household management, mastership, statesmanship and monarchy are the same thing, as we said at the beginning of the treatise; others however maintain that for one man to be another man’s master is contrary to nature [“παρὰ φύσιν”], because it is only convention that makes the one a slave and the other a freeman and there is no difference between them by nature, and that therefore it is unjust, for it is based on force.”

Sadly, this is all Aristotle says about these people who regarded slavery as unjust. He does not mention any specific individuals who held this belief, nor does he say anything about how organized these people were in their opposition to slavery or anything about any other arguments against the justness of slavery that they might have used.

The most parsimonious conclusion, given the paucity of evidence in other sources, is that these people Aristotle mentions who were opposed to slavery were not organized in their opposition in any way and merely held the view that slavery is inherently wrong as a personal philosophical position.

As far as who these people Aristotle mentions were, the only one thing we can say is that the argument he describes them as making, which distinguishes between "nature" and "convention" and holds that people should reject "convention" in favor of living in accordance with "nature" is one that was fairly common in fourth-century BCE Greek philosophical circles, but that is especially closely associated with the philosophical school of Cynicism. Some of the main leaders of this school in Aristotle's time were Diogenes of Sinope, Krates of Thebes, and Hipparchia of Maroneia.

The Cynics were known in antiquity for their radical rejection of basically all social and cultural norms, including the pursuit of wealth, fame, and comfort, instead favoring lives of asceticism. Diogenes, the most famous philosopher of this school, is reputed to have, among other things, slept in a pot in the street, gone around barefoot and worn only a single tattered cloak (which also doubled as his blanket), masturbated in public, and urinated on his detractors. It is conceivable that some Cynics might have argued against slavery on the basis that it is a societal convention that is contrary to nature. This is, however, speculative.

Aristotle himself, of course, only mentions people who thought that slavery was wrong so that he can argue that they are wrong. He immediately goes on to argue that slavery is, in fact, completely natural because some people have naturally submissive, servile dispositions and are therefore clearly intended by nature for superior, more dominant people to enslave. He writes in his Politics 1.1254a, in Rackham's translation:

“Hence whereas the master is merely the slave’s master and does not belong to the slave, the slave is not merely the slave of the master but wholly belongs to the master. These considerations therefore make clear the nature of the slave and his essential quality: one who is a human being belonging by nature not to himself but to another is by nature a slave, and a person is a human being belonging to another if being a man he is an article of property, and an article of property is an instrument for action separable from its owner.”

“But we must next consider whether or not anyone exists who is by nature of this character, and whether it is advantageous and just for anyone to be a slave, or whether on the contrary all slavery is against nature. And it is not difficult either to discern the answer by theory or to learn it empirically. Authority and subordination are conditions not only inevitable but also expedient; in some cases things are marked out from the moment of birth to rule or to be ruled.”

In the centuries after Aristotle, slavery seems to have been generally accepted. Some authors like the Roman Stoic philosopher Seneca the Younger (lived c. 4 BCE – 65 CE) in his Moral Letter to Lucilius 47.1–10, criticized the brutal mistreatment of enslaved people by their masters, but were always careful to clarify that they did not support abolishing slavery altogether.

Even the Greek philosopher Epiktetos of Hierapolis (lived c. 55 – 135 CE), who was enslaved throughout his childhood and young adulthood, in his Discourses 4.1, transcribed by his pupil Arrianos of Nikomedeia, describes enslaved people as yearning for their freedom, but says nothing about any idea that slavery as an institution could be abolished. (In fact, Epiktetos actually seems to mock enslaved people for wanting to be free, insisting that all people are slaves of fortune and that freedom from slavery does a person little to no good unless the person heeds the lessons of philosophy.)

(This answer is continued in the comment below.)