I've trained in Karate, Kenpo, FMA and more.
But it's in Kung Fu that I find many of the most holistic tools for body development
Especially in terms of mobility
This is the Tea Cup Drill
You don't need to hold anything in your hand to do this, but if you do, start out actually holding it and trying to keep the open end uppermost, imagine there's liquid in it. In time loosen the grip Maybe even actually put liquid in it.
Also, don't start in the deep horse stance I'm using, simply stand in a comfortable stance. As competence comes, open the stance and get lower, as you do you'll discover your hips!
Try this today, do intervals of it as movement breaks away from your screen