I would imagine that in the decades before the fall of the Western Roman Empire, Roman subjects would feel unsafe and consider moving to the much more organized and stable Eastern half of the empire, though I may be overestimating the ease of travel at the time, or projecting modern behavior on a population that perhaps didn’t consider exodus as a solution to their problems (instead of isolating in villae). My question still stands though: was there an internal mass migration toward the East?
It isn’t exactly my field (masters in Contemporary History), but I think I remember from my first years lectures that the east Roman Empire was a very different society than the west. They had their own emperor and a different focus. Communication and connections were hard in those times over a large area. Two very different worlds.