One source has been Byzantine tax records ~ but notice that about the 6th century the whole world fell into a triple dip Great Recession like nobody's business. Every standard of civilization was abandoned. The recovery took a long time and even as late as 1790 a full census of the population (see Constitution) was considered quite revolutionary.
One population did manage to get its makeup put under the microscope ~ they were called Huns. When they left China, expelled by the Emperor himself for intense banditry and rapine, they were pretty much Chinese. By the time they got to Europe a couple of centuries later they were White folks.
This happened simply because females do best in settled life but these guys were migratory and continued raiding all across the continent until they reached Europe. This required them to obtain new females on a regular basis.
I’ve read a good bit about the Huns. The Romans and others of the time definitely viewed them as very different physically. Not “white folks” at all.
The Hungarians or Magyars and Bulgars and other steppe groups you are more or less correct.