That sounds like it could be Sidonius Apollinaris. Dozens of his letters survive and provide an interesting view of 5th century Gaul.

I can't find the exact quote but it's something like "I am a Roman, a soldier, and a Frank."

If we look outside the Roman elite, which makes up a very small percentage of the overall population, I would bet money that the vast majority of people hunkered down and stayed put.

Consider two facts for this: most didn't have the financial resources to travel hundreds of miles or give up their day to day livelihood because they needed to put food on the table today. It may not have been subsistence level living but very likely a paycheck to paycheck sort of situation. That handicaps people's ability to make big changes and cut away from their previous lives to start over somewhere new.

Secondly the 5th century was an unstable, chaotic, violent time for most of the provinces. Civil wars and barbarian thugs rampaged up and down and around the Western Empire. Travel was perilous and I imagine robbery and death awaited people who weren't quick enough to get out of the way of those Romans and non-Romans who had nice pointy sharp sticks roaming around and demanding your gold and supplies.

This is not to say that there wasn't movement of people at all. We can look at the Bretons as an example of a group that fled from the barbarian migrations. But even that group already had close ties to the area they went to through trading and kinship and so had a community network to rely on for support (housing, jobs, family, etc.).

I doubt we'll ever know the true numbers of how much of the population moved around but at least on the continent I think people were conditioned to duck and cover after decades and decades (and more decades) of war.