You're basically describing the reverse of Zeno's paradox - the "why bother" paradox.
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.
The rocket equation tells us that if one somehow managed to build a spacecraft containing an infinite power source that weighed a single kilogram, and incorporated the most efficient engine designs known to us (electrostatic ion thrusters), it would still need to be fueled with a reaction mass of xenon equivalent to approximately two billion times the estimated mass of the observable universe to achieve even 10% of the speed of light.
For example, if you had a vehicle weighing 10t empty (but including energy), and a thruster capable of accelerating ions to 1% the speed of light, you'll only need 17t of reaction mass to achieve a speed delta of 1% the speed of light.
This ratio holds up as you scale exhaust velocity and delta V, so the exact same reaction mass would be needed for a 10% light speed delta V using a thruster shooting ions at 10% c. As a rule of thumb, you always need a reaction mass about 170% of the vehicle's dry mass when your thruster exactly matches your target velocity.
Now, to illustrate how critical exhaust velocity is, imagine you make thrusters with exhaust velocities 10x that of the target delta V. In those cases, you'll only need a reaction mass of roughly 10%. However, if your exhaust velocity is 0.1x the target delta V, you'll need a 2200000% reaction mass!
1: https://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation 2: https://en.wikipedia.org/wiki/Ion_thruster#Comparisons 3: https://en.wikipedia.org/wiki/Observable_universe
Not that the former is easy, but the latter is basically impossible. After the former is achieved, we can basically coast on sleep for eternities.
I think if you do launch then you have a clear target to improve on.
And the next one does not have to go to the same destination.
"It has been argued that an interstellar mission that cannot be completed within 50 years should not be started at all. Instead, assuming that a civilization is still on an increasing curve of propulsion system velocity and not yet having reached the limit, the resources should be invested in designing a better propulsion system. This is because a slow spacecraft would probably be passed by another mission sent later with more advanced propulsion (the incessant obsolescence postulate)"
I think this is the error
How do they even know what to improve upon without building it and experiment with it?
Also simulation is not a replacement for real world testing. They should launch stuff into space to figure out if it works and their simulation is making the correct assumptions.
Also I'd imagine these launches are sending back valuable data that they can use to make adjustment and improvement for future engines.
That computer would still feel pretty snappy today for desktop tasks, running dual 2560x1440 screens. Although new AAA games would probably have issues, either low frame rates or not running at all. Most indie games would work just fine.
I think the change will be even less in next 10 years. 2019 hardware will perform just fine in 2029.
Unless something dramatically different pops up, we're in the era of diminishing returns.
For a fixed budget, assuming that Moore's Law still holds (dubious, these days), you should wait until the computation will take 26 months on hardware purchased within that budget [0].
Given that Alpha Centauri is 4 light-years away, this is a fairly bold speculation.
Say it takes 300 years to get to Alpha Centauri. If you launch it you have 150 years to invent a spacecraft that goes twice the speed of the first one. Very likely. If you launch that one it will overtake the other one. Now you have 75 years to find an even faster engine. Launch it and you got 37 years. etc.
Quite likely you can keep doing this until you can reach it in e.g. a year. So why not hold all launches until then....
Does this have a name? Or original source? Did I get this from Futurama?