It's been done several times in fact, though I don't know if it's ever really been a central plot line.
The Revelation Space series has shades of that, but it's mostly background information that doesn't come directly into play. The "Amerikano" generation ships that colonized nearby stars (often less than ideally inhabitable) which were massively outstripped once the "light huggers" which could make the trip in a few years subjective time.
The Sector General has something similar, though with FTL ships replacing the generation ships. I think they find one of the old ships drifting through space, the inhabitants all dead or nearly.