I've heard the date fall of the Roman Empire has been disputed, but if we could discuss what happened to the Roman people throughout that would be helpful. I saw an article in the Atlantic today that put the fall 1,538 years ago. Thanks!
Most of them stayed in the regions were they lived. Of course the population declined in late antiquity and the size of cities were greatly reduced. Rome's population plummeted from 1.000.000 to something like 10.000. There are some cases of romans captured by barbarians who then assimilated into their new environnement. Those considerations are relevant mostly for the western roman empire while the eastern part was still quite wealthy, and survived until 1453.
It has been mentionned on multiple occasions on this subreddit that there was not only one fall of Rome but rather a continuous process of political emfeeblement.
Check this section: http://www.reddit.com/r/AskHistorians/wiki/darkages