The source of Homer's Iliad and Odyssey was ancient Greek oral epic history, where an aoidos would improvise verses that followed the outline of a story, usually to meter and musical accompaniment. Every performer would have their own take on the exact sequence of events and the details, which led to a lot of different variations.
But I've read that it was common to memorize Homer's versions word for word. Did these versions become definitive? When the versions we accept as "standard" appeared, did these variant versions of the narrative disappear?
Brahmin scholars in India would also often memorize the entirety of important Sanskrit texts, including the epic poems attributed to Valmiki and Vyasa. These stories originally come from a similar oral tradition based around live performace, which continued into the early modern period. Do we know whether these performers still sang their own versions from the oral tradition into say, the Mughal or Colonial Era, or was everyone singing the versions by Valkimi and Vyasa?
I can't answer the specifics about the context of the "tellings" of these or other early oral epics - although I suspect that the answer is almost universally that we can't know for want of sources addressing this question. To understand something of the context for how these sources were performed and how much variation there was, we can look at the work of folklorists and ethnographers over the past two centuries. With these, however, observation is often clouded by national aspirations and pride in oral tradition.
I addressed the problem of variation in oral traditions in my book The Folklore of Cornwall: The Oral Tradition of a Celtic Nation (2018 - soon to be issued as a paperback to counter its ridiculously high price!), so I will draw on that here.
There are two general observations that we make. The first is that epics that use some sort of poetic device (rhyme or alliteration, for example) will tend to be more conservative in variations than those that are simply told.
The second observation is that some cultures boast that they have conservative storytellers who repeat their narratives word for word while other cultures boast having storytellers that modify narratives to make them their own. Because early folklorists were hoping to consider collected stories as "documents" that reached back centuries if not millennia, there was a tendency for folklorists to celebrate cultures that claimed to have conservative approaches to narrative transition and to believe storytellers who claimed to be able to repeat narratives exactly as they heard them.
Irish folklorists were particularly prideful of the conservative legacy of their mammoth collection of folktales. Quoting from my book:
In 1945, James Delargy (Séamus Ó Duilearga, 1899–1980), the founding director of the Irish Folklore Commission, presented a classic study of the seanchaithe, the storytellers of Ireland. His essay went a long way towards documenting the expert keepers of tradition who roamed the countryside, telling stories in exchange for room and board. Delargy was pathfinding; folklorists regard his eloquent work as the definitive early discussion about those who conveyed popular stories from one generation to the next. (sources: James H. Delargy (Séamus Ó Duilearga), ‘The Gaelic Story-Teller with some notes on Gaelic Folk-Tales’ (The Sir John Rhŷs Lecture, presented 28 November 1945; published 1946); and see his ‘The Gaelic Story-Teller – No Living Counterpart in Western Christendom’, Ireland of the Welcomes, 1:1 (1952), pp. 2–4. Dundes reproduced his original article in International Folkloristics, pp. 153–76; references to Delargy’s article employ page numbers from this edition. Dundes, p. 157, further recommends additional sources to be considered in the context of storytellers: Peig: The Autobiography of Peig Sayers of the Great Blasket Island, translated by Bryan MacMahon (Syracuse, New York: Syracuse University Press, 1974; originally published in Gaelic in 1938) and her sequel, An Old Woman’s Reflections, translated by Séamus Ennis (Oxford: Oxford University Press, 1962); and Henry Glassie, Passing the Time in Ballymenone: Culture and History of an Ulster Community (Philadelphia: University of Pennsylvania Press, 1982). Gun Herranen provides an example of a Finnish-Swedish storyteller, Berndt Strömberg (1822–1910), discussing the creativity of the tradition bearer. Herranen’s article, ‘The Storyteller’s Repertoire’, appears in Reimund Kvideland and Henning K. Sehmsdorf, editors, Nordic Folklore: Recent Studies (Bloomington: University of Indiana Press, 1989), pp. 63–69.)
A subsequent encyclopaedic study by Georges Denis Zimmermann (1930–) is a twenty-first-century benchmark analysis of the Irish storyteller. He draws on evidence from early material as well as from the later period of classic folklore collection in the nineteenth and early twentieth centuries. With this information, he constructs an elaborate portrait of those who told the stories. The insights of Delargy stand, but Zimmermann adds a comprehensive review of sources. (source: George Denis Zimmermann, The Irish Storyteller (Dublin: Four Courts Press, 2001).
Ó Duilearga tended to accept the claims of the Irish storytellers - that they were faithful to what they had heard. He wrote, for example:
that they 'attained a very high degree of perfection in medieval times to judge from the detritus of the epic literature of the manuscript tradition which has come down to us from the eighth century. But the written saga is but a pale ghost of the tale that once was told, and the personality and polished artistry of the medieval story-teller can only be guessed at by the student of the written word who has not had the privilege of hearing the living voice of the modern reciter of Irish wonder-tales, the democratic descendant of the aristocratic story-teller of a thousand years ago.'
He further suggested that there was ‘remarkable continuity of tradition linking the living oral literature of the Gaelic West… with that of the manuscript tradition in an unbroken chain of over 1,200 years’
Zimmermann called this into question, suggesting that even if this was the goal, it would have been impossible to be completely faithful to a narrative that someone had heard and then repeated. He suggested that Irish storytellers took pride in repeating the stories with a great deal of fidelity and that they may have even believed that they were repeating the exact words, but that was not likely.
As indicated, other cultures do not claim this sort of faithfulness in the telling of the stories. The Cornish drolltellers - the professional storytellers of Cornwall (the object of my study) were repeatedly described as boasting that they changed their stories to suit the moment. Alan Dundes, in his 1964 classic study of North American storytellers made a similar observation about Native American storytellers; in his case, he was advancing an approach to folklore that depended on widespread variation in the retelling. He also encountered some cultures that were more conservatives in this regard than others, but Dundes simply noted them and then set them aside. (see his The Morphology of North American Indian Folktales (Helsinki: FF Communications, 195, 1964.)
The problem I ended up dissecting was that we had both storytellers who claimed to reside on one or the other end of the spectrum - fidelity or creativity; and we had folklorists/ethnographers who sought one of the other in the cultures they considered. There can be, consequently, two layers of subjectivity among those who attempt to answer the question you raise: there are those (storytellers and scholars) who want the sources to be as conservative as possible, and there are those (storytellers and scholars) who boast of as much artistic variation as possible.
Unraveling this dynamic when it comes to the obscure context of ancient sources is maddening. While we can probably believe those who claim to vary the stories they heard, we need to approach those who claim strict faithfulness to the text with some amount of source criticism. And yet, there is no question that when a culture demands fidelity in the transmission of an epic narrative, generations of storytellers will do everything that they can to deliver on that expectation.
To answer your question, it is consequently important to first attempt to understand what a given ancient culture seemed to be demanding of its storytellers. If there is a tradition that they were memorizing the text as they heard it, then we can accept that this, at least, was their goal. By wary, however, of scholars who want this to be the case and claim to see evidence of that level of conservatism in ancient cultures!
Given all of this, I can at least provide a framework with which to evaluate your question:
Was the source reliant on some poetical devices? If yes, there is likely to be a level of conservatism.
Did the ancient culture in question seemed to boast of the conservatism of the process of transmission from one storyteller to the next? If yes, there is likely to be a level of conservatism.
Do modern scholars see a level of conservatism without an affirmative on at least one of the above two points? Then view what they claim with skepticism. Some cultures, after all, celebrate change, and we must assume that this was a possibility in the ancient world.