Peasant migration? In the Middle Ages? It's more likely than you think. Of course, we must lead with the caveat that the Middle Ages span a long time over a large number of places, so what goes for England in 1451 may not be what went for Aquitaine in 922 - or even for England in 801. Here's some previous links for your perusal:

• u/Asinus_Docet provides an overview of some relevant factors, written for that OP's young daughter and therefore simplified, but a good breakdown of what a peasant might have to consider;
• and u/sunagainstgold covers a common migration pattern, that of village dwellers up and going to the cities.

And as always, if anyone wants to put in an answer of their own, please don't let this post stop you!