hi! you might be interested in these threads

• sheet music How did people hear new music before radio and record players? (there are actually several posts on the history of musical notation if you're interested in that topic)

• player pianos How did musicians respond to the player piano when it was first invented?

I cannot help with popular music of the past, and cannot tell you if there is a certain most well traveled piece of music. However...

For most of history (and before it), music is and has been communicated via oral traditions. That is, I sing/play something and you pick up... Now you know it, too. Then this music travels just like languages and stories (that is pretty complicated, and way out of my league). Music changes during its travels, just like languages and stories. You might want to also ask this question to linguistics, anthropologists, etc.

Modern Western music notation comes from one form of notation created to address the problem of spreading music in a controlled way. During the 9th century, forms of neumatic notation were developed. These were meant to be aides for people to remember music they had learned before. There were some standardization efforts during the so called Carolingian Renaissance, they tried to have musicians all across Europe singing the same chant (there were different versions, and different chanting traditions).

Gregorian chant was notated and it traveled all a lot. Musicians were trained in singing it AND reading the notations to be able to sing it properly. We aren't 100% sure how it's supposed to sound, even after all those attempts at standardization, a lot of details are lost in time (and in translation, and in transcription).

Now, that's a repertoire, not a single tune.

The Follies of Spain come to mind. It's a musical idea from at least the 16th century, it traveled all over Europe and managed to cross the Atlantic, it is found even in music from the 20th century. Here's an incomplete list of examples where we find this "tune."

Here's a banging performance of a Sonata by Vivaldi (a venetian composer) that uses this musical idea. Compare it to this interpretation of a sarabande (a dance from a suite) composed by Handel (a German composer that managed to conquer England). That music was composed relatively close in time.

Now listen to this music, by a Mexican composer around 1931, about 200 years after the Vivaldi. Here's more music using the same idea, this was composed by a super famous Russian musician (who was living in the US) about a year before the previous 1931 example.

There are PLENTY of settings for that musical idea.

Musicians traveled before recording technology. Mozart, for example, could be considered TODAY as a very well traveled person. Notation allowed music to travel in an easier way, even without its creator or without the people who know it best.

I listed a sarabande in the previous examples. Well, that type of dance was apparently invented in the New World, during the early 16th century (maybe earlier). And it was taken to Europe by the Spanish, where it developed into some rather different things. At first it was considered "indecent" in Europe, but it managed to become popular, and then it was not danced at all (it became more of a "form" for musicians to work with). It survived into the 19th and early 20th century (I guess we might still find it in the wild these days). The Follies of Spain were frequently set as a sarabande.