A lot of people think it looks like body-swapping, but that's just a superficial view of it. There's likely a reason why it's officially being called "possession" rather than "swapping" despite being described as swapping.
Chirorian's
theory is that while they were sleeping, Mokou's dream soul possessed Reimu's body, and for the time being Mokou's real body would have been without that dream soul, but there'd actually be two Mokous. Apparently the ending suggests that maybe the swap wasn't perfect, so Reimu didn't invade Mokou's body at the time. Anyways, they're unconvinced that it's actually a body-swapping thing.
If you read WaHH 29 and 35 you'll get an inkling of what might be happening, but here are my thoughts. It's still kind of unclear though.
When Sumireko first started coming to Gensokyo, somebody sleeping would have their own dream soul displaced by Sumireko's, and it would just wander around afterwards. Eventually she "displaces" entire human villagers while they're sleeping, but they seem to kind of stop existing while Sumireko takes their place, and they're returned in their entirety in the same place they were sleeping. Basically my take on this is that their dream soul is displaced from their body by Sumireko's own dream soul, and the physical space their body takes up is used by Sumireko to sustain her own form. Meanwhile, the villager's dream soul just kind of wanders in the meantime, and Sumireko's actual body is left without its own dream soul.
Sumireko's location in Gensokyo doesn't seem to depend on the villager's, which is a major difference from this new possession, but the concept is similar: Mokou's dream soul possessing Reimu displaces hers from her body, and the space Reimu's body takes up is used by Mokou to become Mokou. Mokou's real body is still where it was, without her dream soul, and Reimu's dream soul is off wandering. This doubly makes sense if you suppose that a
perfect possession would have the two people swap entirely, since they'd essentially be swapping dream souls, and the bodies (which are similar) change form to become the possessor's.
Surely a lot more confusing than lel body-swap, but yeah.