Wasn't willing to write an answer to the q, as OP clearly has no idea for how much blood and suffering he subscribes to those guys for a period from 100'000 years to 10-20 million years.
So as the approach basically condemned by almost everybody unconditionally, and what should take place is worse than it by an order of magnitude. For that reason, I won't dive into gruesome details.
The reason I decided to write an answer because current answers and comments are soo close to naming it, but they didn't. Another reason is its good example of how pointless it is with genetic modifications(which more humane way) or without it, in a hard way as OP wants, I guess, as OP didn't clarify and probably long forgotten he asked it on WB.
End of intro, real deal now.
Basically, it is the same selection breeding process we use for dogs, cats, domestic animals, but for humans, with or without some belief system attached to it.
- and it requires it as they need to eliminate(kill) people who do not fit, 10x This is Spaaartaaaaa way, be like Aztec's guys who did human sacrifices out of the belief that if they do not it then a sky will fall. It could have some sense for their way of life and situation, as population regulatory mechanism, so it, not total nonsense, words are wrong but. But with eugenics on our planet is a road to nowhere, but on an OP's planet/idea it is the way to keep going.
First success stories dogs, cats, especially dogs as it is easier to crossbreed them there are many different kinds of dogs, few thousands to be more specific, and they have different builds and exteriors, from tiny tiny things to quite big ones.
Sure we breed them to be dogs, so they still dogs, after 10000 years. But if one has quite a pool of genes and incentives to do so it possible to shape them quite fast, through selective breeding to a variety of dogs. And if you look at well-known breeds of dogs, how did they look a hundred or two hundred years ago you can see the differences easily, they can be quite full of changes - no wings obiviously, but it just 100 years.
So a thing everyone forgot about humans, in their answers or comments, me included - is The Brains
We aren't animals, who need to evolve mindlessly, they do not have enough brains to judge things, but humans do.
The bad news however, potential speed of changes in dogs is about 60 times faster than for humans - 1.5 years dogs can be considered mature, and it brings 4-6 pups each year. Humans 15 years, once 2 years, 1 kid typically, soo even cows are 10 times faster(3y maturity, 1 calf, each year)
JDługosz in his answer correctly mentions that diversity of genes will be a problem, indeed, but he decided to be short and didn't explain, so I will do it.
Before humans knew what DNA is before we knew how to make GMO - the plant selection process was quite long inaction, and actually was or is modeling process of inheriting properties and indirectly pointing on existence of genes for guys back in the days who tried to work out the logic behind the process.
Due to different mutations over millions of years, plants do(or did in some cases) have enough of gene diversity, and some combinations of those collected together do create breeds of apples we know today, avocados, bananas, that thing full of seeds we make breed from, etc.
Significant improvements were achieved by the selection process, randomly(not exactly) taking some combinations out of many many possible ones for that gene pool and working on improving or dismissing results.
Avocado is quite a prominent example. Random, meaning typical avocado guy/tree tastes like trash, out of the whole population of avocados trees only some taste good, meaning there is a big enough pool of genes, but only specific combination tastes good enough to be sold. And what is sold are products of clones of a specific member of that avocado population, and if you breed one in a natural way(seed) from those there is a very slim slim chance(0) to get good tasting one.
- meaning - if avocado trees would be humans, and we would be vampires, then nice tasting ones would be, maybe, red hairs with a good smell in their armpits.
Stranded spaceship indeed may have no diversity of genes, and they will become even poorer in that aspect each next generation, well-known problem for generation ships, discussed here multiple times, and it is a bridge to cross and not to do die out, but Australian tribes do show it possible.
But if they get their numbers up, then they do not have to rely on mother nature to enrich them with mutations, drink more C2H5OH and it increases the probability of mistakes in chromosome divisions and mutations. So the rate of mutations can be increased by behavior, so as finding more potent sources and ways, meaning it does not necessarily has to be radiation, mundane chemicals can do that as well.
The most gruesome part of all that will be the selection process, it far beyond the typical imagination of a human ... what? What did u say? Ahh - nazi, cannibals, pirates, mengele, italian mafia, narco cartels - no no no, those are innocent kids.
I do believe op just innocently didn't think about those things through, and was captivated by Disney, but the truth behind that is ...
But things are worse than, how can it be - do you ask, well suffering for meaningful cause can be justified, but this one, ...
We really have to be grateful to our ancestors, they did have a hard life, by those ancestors I mean even those well in the past, maybe even not homo erectus.
The human species got two things right - brains and tools and through their efforts and ingenuity and hardships and application of those two things - we are where we are today. Yes, there is a long road ahead as well, but because of applying brains to the usage of tools and ideas I can type my answer in the comfort of my home for you to read, instead of being in a cave or on a tree thinking which predator will try to eat me today.
Thank you to all our ancestors! Hallelujah! Your collective work made today possible! Hallelujah!
Special thanks, from me personally to you, NoName guy, who made stone knife and equipped humanity with Claw's, and to you who made Atlatl - you guys are great, you did the life of humans easier and survivial possible. It not necessarily was a big step for you guys, but it was a jump into the sky for humanity.
OP what do you think your stranded descendants will tell about you?
Use technology, few thousand years and you will make a pancake out of that planet if it will be what's your desire.
P.S. People feel themselves quite well on the water this is one of the examples https://en.wikipedia.org/wiki/Sama-Bajau