Scholars seem to disagree, with some saying that most people never traveled further than ten miles from their birthplace, and other people pointing to the popularity of pilgrimage and other examples of medieval migration.
What I would say is that because most migration tended to be local and regional rather than long-distance, it would be unlikely that you’d see commoners moving to the North from Dorne. Rather, I think you’d see a lot of internal migration by Northerners and external migration from the northern Riverlands and the Vale.