It's been done.
I mean, time dilation in general, and a fast ship catching up with a slow ship in particular.
I would guess 50 years ago-ish practically every variation had been done. Tau Zero was a notable example.
I don't know when the genre started, but it's easy to forget or not be aware than Einstein came out with special relativity almost the exact same time as the Wright brothers were getting their plane going.
Technology has taken a long time to catch up to physics.
This didn’t, of course, deter their crews from wanting to fight the battles anyway. They were trained, they were ready, they’d had a couple of thousand years’ sleep, they’d come a long way to do a tough job and, by Zarquon, they were going to do it.
This was when the first major Muddles of Galactic history set in, with battles continually reerupting centuries after the issues they had been fought over had supposedly been settled. However, these muddles were as nothing to the ones which historians had to try and unravel once time-travel was discovered and battles started preempting hundreds of years before the issues even arose. When the Infinite Improbability Drive arrived and whole planets started unexpectedly turning into banana fruitcake, the great history faculty of the University of MaxiMegalon finally gave up, closed itself down and surrendered its buildings to the rapidly growing joint faculty of Divinity and Water Polo, which had been after them for years.
Which is all very well, of course, but it almost certainly means that no one will ever know for sure where, for instance, the Grebulons came from, or exactly what it was they wanted. And this is a pity because, if anybody had known anything about them, it is just possible that a most terrible catastrophe would have been averted — or, at least, would have had to find a different way to happen.
It leads to lots of really interesting and fun sci-fi that somehow hasn't really been dug into sufficiently yet along the same lines as the Twin Paradox [1]. We send out a colonization ship traveling at some relatively fixed acceleration rate that (from the perspective of those on the ship) will arrive at their uninhabited destination millions of light years away in about 40 years.
Then thousands of years later (from the perspective of those on the planet) we make a breakthrough enabling us to develop ships with several times the acceleration, and send another ship to the same destination. The first ship we sent out may arrive only to discover that the planet is at a late stage of development having been colonized by the second ship of humans tens of thousands of years ago.
I mean imagine landing and the humans on a planet are no longer even "human" in the sense that you knew - technological and biological evolution alike leaving you as a relic from the past. Though even for those who do not travel. Imagine yourself being one of the people on said planet. From your perspective, such relics of ancient past arriving and departing, would also be a regular occurrence.
The entire idea of time starts to get really strange. Also the above is not entirely just theoretical. For instance the exact same phenomena that bends time every which way, also causes particles in particle accelerators, traveling at relativistic rates relative to their observers, to decay much slower than they would otherwise - which is pretty neat and handy!
[1] - https://en.wikipedia.org/wiki/Twin_paradox