If it's a period of collapse, perhaps complete collapse, humanity will have lost an opportunity by delaying the launch.
https://en.wikipedia.org/wiki/Interstellar_travel#Wait_calcu...
An interesting example was "Interstellar Travel - The Wait Calculation and the Incentive Trap of Progress"¹, which somehow isn't on libgen [yet?].
1. http://www.jbis.org.uk/paper.php?p=2006.59.239 - The incentive trap of linking to a £5 download for fifteen year old paper is another paper entirely.
https://www.researchgate.net/publication/260275150_Interstel...
(edit: I presume the downvotes are for linking to Researchgate? I know they're not held in high regard like Arxiv, they're more like a sleazy Linkedin for academics, but if there's valuable content there why not use it?)
For example, in the Honor Harrington books by David Weber, the Manticore system colonists traveled to their target a couple hundred years in hibernation. But as it was likely propulsion technology will advance before they reach their destination, they also left part of their money in a trust fund back on Earth, to make sure the Manticore system won't get squatted in the meantime.
This proved to be a sound idea, as practical FTL drives have been invented after the colony ship has departed and they have been greeted by the Manticore Trust navy upon arrival, that has secured the system a couple years earlier, arriving by the newfangled hyper drive. :)
The lore is quite amusing and poignant. Many of them met a sad end in one way or another, reminiscent of Fallout.
Then Country Y invests in new technology and starts building their new ship in, say, 100 years.
It's possible and likely that Country Y's starship is more capable and faster than Country X's ship as they would have waited for and capitalized on scientific breakthroughs that make interstellar travel better.