I don't know as much about the Middle Ages, but in Ancient Rome people could be quite mobile. There was a lot of seasonal migration: tens of thousands of people would leave their farms in the summer and come to Rome to work as porters in the harbor or as construction workers, to make a little extra money. Traders would sail all over the Mediterranean, bringing goods from as far as India. People traveled for education too, to universities in Rhodes, Greece and Alexandria, Egypt. Basically, if you were a citizen, you could go anywhere. Also, funerary inscriptions and isotope analysis of teeth and skeletons have revealed thousands of people of foreign origin living in Rome. The details of immigration in Rome are sketchy; most of the evidence we have is funerary in nature, and frequently remains were repatriated to their place of origin. But if you want to learn more about the subject I'd recommend Moving Romans by Laurens Tacoma, which focuses on the early empire, a time of peace when lots of people could safely move throughout the highly connected Roman territory.