Did expanding empires in the ancient and medieval world bring about large or medium scale migrations? So the possibilities could be:
They did not, a lot of mixing between different people happened only in prehistory when people freely moved around, either they weren't bound to any part of land by land ownership nor were they nationals of any state.
An example in the modern world of this could be the colonies of European empires in the 17-20th century, it as only the new world that was populated, otherwise people did not mass migrate to newer colones of their empires, even if they did, they hardly mixed with the locals(an exception would be some parts of Latin America). Today there's no mass migration, with visas and permits, there's only small scale migration to more developed countries.
Did the same happen older times? For example:
1. Roman empire, did moving out to newer colonies offer any incentives? Did people move out in SIGNIFICANT numbers? Now I dont know what "significant" could be, but are there any records? Or did people just stay for a few years in conquered lands and then come back?
2. Nordic tribes moving south, but again we they were tribes, not settled empires.

So the question is, did expanding empires, say the Byzantine empire or Western Holy Roman empire, or even kingdoms which came later on bring about major demographic changes. Or has this process of significant demographic changes stopped long back?