This is a fairly common trope in SciFi settings which have Generation Ships.
Some time after the ship gets launched (centuries, even), FTL gets developed. Then they either arrive and find a civilization at their destination, or they are considered to be off-limits and not to be messed with, depending on the setting.
That assumes such improvements are even possible.
At least for non-magical propulsion, we don't really need "faster" engines. Take whatever engine you want, let's say it can accelerate at a constant 1g. In less than 4 years(from an observer standpoint), you are at 97% of lightspeed. From your point of view, that would take less than two years.
If you can have an engine that can accelerate at a constant 1g for years (fueling it is left as an exercise for the reader), then you are all set. You can go anywhere in the galaxy inside a human lifespan – of course, assuming you don't care for the people left behind.
If you want to accelerate at more than 1g, there may be physiological issues.