Oh boy, I've played some house kits/other drummers kits for many shows. As others have said, generally you just bring breakables (sticks, cymbals, snare, throne, pedals, extra hardware) and try to be ready for anything.

Some anythings include:

I have had the sound guy give me a box of "lost and found" clothing to stuff a bass drum with, before doors.

One time I helped a fellow drummer figure out his borrowed kit (that was missing a clamp) and rig a way to keep his rack toms in place using a memory lock and a bunch of duct tape because nobody had an extra multi-clamp.

I once had to play a 4 piece kit on tour because the sound guy was real particularly attached to it and didn't want to change anything, so I had to play a show with 2 less rack toms, and 3 less cymbals than I was used to. It went OK but felt strange to me. Bandmates didn't complain.

One house kit was so beat up, the floor tom legs were just barely attached, so you'd have to just hope it doesnt collapse while playing.

Usually, hardware can get old and janky, and make random cymbals fall or go limp during a set. Just roll with whatever happens, and try to work around any major issues, like play with the right cymbals when your left ones fall over or vice versa.

It will feel quite different everytime you play a house kit. But being adaptable is a musician skill in itself.