The Eastern Empire/Byzantium was more or less surrounded by the turks. In its latter days, the empire was a diminishing island in an Ottoman sea. The western empire some places, like Britannia, it more or less disappeared overnight, others, it just withered away. I suppose there was a lot of internal displacements in such turbulent times, but mass migration from west to east, seems unlikely. To me, that is :)

Even by 300 AD roman peasants were bound to their landlord, and forbidden by law to leave the manor. Thus a large part of the population was unable to migrate. In order to maintain the empire's armies, and to fund the frequently recurring civil wars, the tax burden on roman citizens had become unbearingly heavy. This combined with the suppressive religious policies of a now Christian empire, made many people welcome barbarian rule. (As happened when roman North-Africa came under muslim-arab rule two centuries later. Also, these were troublesome times, and travelling for long distances could be very hazardous, so my guess is that there might have been migration from areas close to the eastern part of the empire, and perhaps some wealthy citizens (merchants and others, whose wealth was not tied to the land), but all in all, I'd go for very limited migration.