This isn't really an answer to your question, but there has been a general recognition in the field that the "decline" of the Roman Empire was not obvious to Ancient Romans like it is to us. If you look at Rome at the beginning of the fifth century, things are great! Egypt and North Africa are growing lots of grain. Taxes are being collected annually. It's a Christian Empire, ordained by God and claimed in the blood of the martyrs (or at least that's how fifth-century poet Prudentius presents it.

I would say that Ancient Romans didn't think of their Empire as being in decline. Augustine, who lived through the sack of Rome in 410, seems relatively unaffected by it. Even Justinian in the 6th century still has a sense that the Roman Empire is the whole Mediterranean.

This is not to suggest that people didn't flee the cities during moments of actual destruction. For more insight, check out Chris Wickham's The Inheritance of Rome.

As an additional related question, did Byzantine rulers take any measures to offer asylum to imperiled Western Romans?