never mind.
the majority of transfers between boarding schools are due to getting kicked out of the first one. full-pay kids, usually connected in some way. it’s fine.
(most of) the only other times you see it is indeed for sports. both schools kinda come to an agreement of sorts that the kid is not going to get what they need at the first school and will suffer moving to the next level, usually college, and everybody on all sides works to make it work out. often you get a repeat year out of it to the benefit of the new school and their team. again, it’s the grossest of favoritism but also the way this little world works.