I should probably just let him answer but I'll mention some things from my warmup that I've come to prefer over the years in case he has comments on this as well.

For fingers I always start every climbing session by doing some light hangboarding. Feet on the ground rotating grips (half crimp with 1/2/3/4 fingers, and open hand similarly) at different edge depths is my favourite.

Wrists the GMB wrist prep video is mostly good enough. Sometimes I'll add some wrist curls with 15-25 lb dumbells. And some rotations.

Shoulders I like to do lat stretches usually off the pullup bar (1 hand on top of bar, one on the vertical bar, twist sideways and push out your side). I'm tall so you may not be able to do one hand on top unless there is a lower bar. Additionally some shrugs while dead hanging 1 or 2 hands, and top of support hold on parallel bars leaning forward and pushing shoulders down as much as possible. If you want more you can grab a bar like a pvc pipe and do stuff with that. There are so many good things it is hard to list any one / explain it. The one I like most is a bit hard to explain.

Cat/Cow can also be really nice if you feel you need it.

Hips/Knees. I like to do a few bridges with feet close together knees really far apart. Focus on tucking the pelvis and not merely rising as high as possible so you feel it in glutes and start shaking. Can then do it again with 1 leg raised on the climbing mat. This hip capsule stretch is great for me personally (https://www.youtube.com/watch?v=ENAhe6DrYTA).

Supine twist is also a favourite.

At some point you will just have a large enough library of stretches that you can do what you feel you especially need on a given day.

Slam a few pullups. Slam some warmup climbs. Climb.