I was originally going to post this as a reply, but it got pretty long and I felt it might as well be its own post.
While we're thinking about Aesop, a second cycle of animal fables were also making the rounds in Europe and were also quite popular, from the Middle Ages to the 19th century. In Europe, the were largely known as the Fables of Bidpai. Looking at their history might provide an interesting comparand to illustrate how these tales moved and circulated from place to place.
The fables are traceable back to India, where they were known as the Pañcatantra. They date as far back as the 3rd century BCE, roughly corresponding to the time of Alexander the Great. By 570 CE, they had been translated into Pahlavi Persian and were well-established in the court literature of the Sassanian Empire under the title Kalilag o Damnag, the names of the two jackals who are the central characters of the cycle, from their Sanskrit originals Karataka and Damanaka. Following the Arab conquest, the Zoroastrian convert Ibn al-Muqaffaʿ (d. 756) translated these fables from Pahlavi into literary Arabic as Kalilah wa-Dimnah. This edition of the text infused the ancient stories with one of the highest examples of literary Arabic style in existence (they are still studied today as examples of first-class writing) and introduced a number of new tales that soften the Buddhist element of the original stories and introduce some of his local Persian-Islamic sensibilities (although it has to be said that he is known as an excellent translator and did not shy away from controversial subjects, so we cannot accuse him of bowdlerizing the text). At the same time, there was a second original translation from the Pahlavi into Syriac, again around 550 CE, but this edition was left childless.
From the Arabic text of Ibn al-Muqaffaʿ, we have a Greek edition by Symeon Seth (c. 1080), which was then translated into Old Slavonic and Italian (1583), a modern Persian edition (c. 1120) that became the basis for the Ottoman Turkish translation, which was then translated into French by Galland (1724) and Cardonne (1778), a Hebrew edition (c. 1270) that was translated by John of Capua into Latin as the Directorium, which then became the basis for German (1480), Spanish (1493), and Italian (1552) branches. The German branch was rendered into Dutch and Danish in the 17th century, the Spanish was translated into Italian as Discorsi degli animali in 1548, and the Italian 1552 branch was translated into English by Sir Thomas North in 1570 as The Morall Philosophie of Doni: drawne out of the auncient writers. Many of these editions took the name Fables of Bidpai, after the Brahman philosopher Baidaba (Bidpai, Pilpai) who is the advisor to the new Indian monarch after Alexander's deposition of Porus (Fur), back in the 3rd century BCE. Finally—gotta love the Andalusian element—the Arabic edition was also directly translated into Spanish (1251), and then translated back into Latin by Raimond de Beziers. Meanwhile, a Latin poetic imitation of the cycle, Baldo's Alter Æsopus, was circulating as well.
Another case study of animal stories in circulation is the story of "The Case of the Animals versus Man," a long epistle-satire written in the style of a fable by the Brethren of Purity (Ikhwan al-Safaʾ) in the 970s. The text was copied from Arabic into Latin and Hebrew, one by Rabbi Joel and the other by Rabbi Jacob ben Elazar around 1240. Then, in 1316, it was translated again into Hebrew by Kalonymos ben Kalonymos, who incidentally also translates "Kalilah wa-Dimnah" and the works of Averroës into the same language. This Hebrew edition was printed in Warsaw in 1879, from which Yiddish and German translations were published. In addition, the Arabic had also been translated into Urdu, and then at the insistence of Abraham Lockett, a British official in colonial India, the Urdu text was translated into English! Thus, we have two totally different branches of the text meeting in Northern Europe, one through medieval Spain, one through India.
Given this daunting history of translation and re-translation, one can imagine that these tales might have been experienced in successive waves by any particular populace. Perhaps a man living in 14th century Provence would have heard the Fables of Bidpai orally from a Spanish Jew, who had read the Hebrew; perhaps his grandson read a Latin translation while at university; perhaps his grandson's grandson goes to Venice and hears the same stories in Italian; and perhaps, generations later, a descendant of theirs, seized by the wave of Egyptomania that swept the country following Napoleon's invasion, picks up a new French translation of the Ottoman Turkish Anwari Suhaili, "Lights of Canopus", and discovers the whole thing all over again, except where the word "God" appeared in the old Latin edition nobody read anymore, will now read "Allah" and imagine the stories taking place in the exotic locales of the East. I'm getting romantic here, but I think what these case studies might help us with is imagining how the stories of Aesop, Bidpai, and countless other bodies of lore that exist both orally and in written form could have been experienced and transmitted by people in the past.
Keith-Falconer, I.G.N. Kalilah and Dimnah, or, The Fables of Bidpai : An English Translation of the Later Syriac Version after the Text Originally Edited by William Wright, with Critical Notes and Variant Readings. Amsterdam: Philo Press, 1970
Goodman, Lenn E. “Reading The Case of the Animals Versus Man: Fable and Philosophy in the Essays of the Ikhwān al-Ṣafāʾ.” In Epistles of the Brethren of Purity: the Ikhwān al-Ṣafāʾ and their Rasāʾil, edited by Nader El-Bizri, 248–274. Oxford; New York: Oxford University Press, 2008