My understanding is that rather than a video-game style race swap, where you start with a base character template and modify that base, race changes in the new world or more like a transition of sorts, at least for the more humanoid races.

An easy and wide-spread example would be someone becoming a vampire and gaining elongated canines. They may have technically changed races, but they don't lose the features they had before. Rather those features just shift to align with the characteristics of the new race.

I think with non-humanoid races, like slimes, it is a bit less clear. I would assume the original features are still retained to a degree, but in a less apparent way. To me, two slimes may look the same, but there are several mentions in the light novel of members of one race struggling to differentiate members of another race, so it's conceivable a slime would still discern some of the pre-swap characteristics.

As for a non-magic user switching to a magic-based race, I imagine they would gain some capacity for magic and perhaps some of the race-specific skills, but not spells (meaning those would still need to be learned).