It seems that there are three main contributors to speciation: separation, small gene pools and lots of time. Environmental pressures factor as well, since it's the environment that historically provided the main selection pressure, but humans have been subborning that for centuries now. Small gene pools are readily resolved by having large colony complements - hundreds of thousands at least, preferably millions. You're clearly wanting this to last, so we're not going to get around the time factor. That leaves separation as the best point of attack.
The first solution that springs to mind is cross-polination. If there is enough interbreeding between colonies then any deviation from baseline human would eventually spread through the entire species, evolving the species as a whole rather than branching into incompatible sub-species. A constant flow of emigration between wide-flung colonies may be costly, but the alternative is a little terrifying: war between insular worlds, each of which believes themselves to be the one true human race and all others are subhuman animals.
Yeah, bit dark there. But common enough in SF to be a trope we can leverage.
For this to last though, it needs to be deeply embedded. Not just legally, not merely logically established plans, this needs to be so deeply embedded in human society and morality that it is never questioned. You're going to have to program your colonial societies to desire miscegenation (which is a scary word, usually perjorative, but entirely applicable here) at a deep level. Insularity and racial pride must be crushed ruthlessly, and children will have to be indoctrinated towards your cultural norm.
Yeah, lots of scary words in that paragraph. Because that's what you're going to need to make this work across deep time. But your goal is pure, so the methods don't matter... right? Anything goes as long as your goals are righteous.
So let's start with inter-system colonial law. Every colony is required to accept a certain number of unmarried, fertile immigrants of mixed genders each year (or whatever time period fits your FTL system best), and to supply an equivalent number of suitable emigrants. Failure to do so will be met with varying degrees of censure from trade embargoes all the way up to eventual dissolution of the colony and redistribution of the colonists to other worlds. Worlds that resist will be cleared of all human life one way or another, and may be restarted with a fresh stock of willing colonists drawn from a variety of other worlds.
The reasons behind these laws are of course well publicised, so that we know that it's our duty to ensure the health of Humanity's future. This is where it starts to become a moral imperative, a duty to our descendants, to the trillions of lives yet to be born. This gives us licence to do some things that might be considered a little... worrying to some. As long as we're doing it for the "right" reasons though we are clear in our conscience.
Having established a moral imperative, we can proceed - clear of conscience - to indoctrination. All news media outlets will be encouraged (by threat of legal censure if necessary) to spin their reporting subtly in the direction of xenophilia. Insular groups are to be vilified, immigrants and mixed groups are to be portrayed as favorably as possible. Works of fiction will be lauded highest when portraying insular groups as ugly, villainous social cancers and 'xenos' as desirable heroes and positive influences. When anyone espouses any form of exclusion they should be destroyed thoroughly in the media, and we'll allow any lies to be told about them if it serves this purpose. Meanwhile we'll encourage active suppression of negative reporting or portrayal of xenos as it could harm our lofty goal of galactic integration. We're not monsters of course, we're not going to make it illegal to marry people from your own colony or racial group, but by normalizing mixing on all media we can at least make it mildly distasteful. If we get this part right we can make it eminently desirable to marry outside your colonial group.
And finally, let's address children. They're not that difficult to indoctrinate with the right methods, and they'll carry that indoctrination into their adult lives. It starts easy, making minor changes to stories and myths to normalize xenophilia and demonize insularity. Rewrite the children's stories of monsters and princesses and all that tripe. Princesses (and princes) are always from exotic worlds, monsters and evil antagonists come from terrible, insular coloniss. Children's visual programming should be gradually filled with heroic xenos, evil insulars, etc. The bad guys are always from some colony that cut contact with the rest of the galaxy, and the good guys are teams made up of the best representatives of multiple worlds. Young adult programming will have endless examples of troubled same-world relationships, while the only perfect relationships will be between people from different colonies.
We don't even have to be particularly gentle with it. Anyone who objects is nominating themselves to be the bad guy in our society, and can be used as object lessons in why you don't fight the narrative.
Yeah, I'm a terrible person. But really, how does one program a society otherwise? If you let people go their own way they might not do what you want them to do. Social programming is the only way you're going to get this to work in the long term, and it's the long term you're really worried about.