A good example is the spread of wheat cultivation across Europe. We can get fairly accurate reads on this because of DNA sequencing of wheat and it was a new crop. There is cultivated wheat at the bottom of the English Channel, meaning it was farmed there before the English Chanel was flooded by the North American ice sheet melting.
Another example is the from the Bronze Age (>3300 years ago). There was a vast trade network at this time that we have direct evidence for [1], being a ship carrying goods from all across the known world from modern day Afghanistan to Europe.
The Bronze Age spectacularly and suddenly ended, which is a whole other story. It's still speculated as to what caused this. One interesting aspect are the so caled "Sea Peoples", that were directly documented (and drawn) by the Egyptians at the time. There's speculation that the "Sea Peoples" themselves were refugees from some other disaster. This too hints at great mobility that so many people could migrate so far.
Trade and human movement in historic times was way more extensive and common than you might otherwise assume.
[1]: https://medium.com/teatime-history/bronze-age-shipwreck-reve...