Procopius of Caesarea (500 - 565 AD)

Taking about the Migration, there is more indications that there was Emigration from the East Roman Empire to the Lands of the Barbarians:

Consequently there was a constant stream of emigration not only to the land of the barbarians but to places farthest remote from the Romans;and in every country and city one could see crowds of foreigners. For in order to escape persecution, each would lightly exchange his native land for another, as if his own country had been taken by an enemy.

Taken from:

Procopius of Caesarea:The Secret History


Or you can see it here on page 38: