I’m doing rotational play and I’m wondering what will happen if I switch household when their kids are still at the boarding school? Will they stay there until I come back to their household and turn their age on? Or will they just come back home right away when I switch.

Technically they should stay at boarding school don't really know never used the boarding schools those are nearly as broken as the Free Vacation.

I've only ever sent one sim to Boarding School. Had no problems with him staying there in between household switches nor with getting him back again after having rotated out to play other households in rotations in that world when it was really time for his school career to be over. But I do not allow "my" households to age while playing elsewhere in town so I am certain that factor helped keep him safe and have no idea what would have happened if I had accidentally or on purpose allowed aging to stay on while the household was inactive or if I had rotated gameplay out to a different homeworld (this was long ago before I started doing that routinely).

That's all I can offer from personal experience. Besides nothing bad happening programmatically, I did not find the experience to be satisfying to me personally and will likely not use it again with or without the risks indicated.

Hmmm Switching homeworld part sounds quite risky...