I don't ride English obviously but I read this thread the other night, I have been thinking about it and will offer what was told to me, wrong or right.
Hopefully someone will correct me or give some insight on what I was told because I don't completely understand it myself.
Doing stirrupless work, but at a standstill, I was asked to move my legs straight out away from the horses sides without losing my upper body position(keeping my core engaged) I found I could only do about 2-4 "leg lifts" without weakening in engaging my core and having to use my whole body to help me move my legs in the lifts and it caused some major cramping in the muscles(no idea what they are called) that run along the side of the butt and hip. I was told leg lifts on the floor would help strengthen those muscles. I am pretty sure that strengthening those muscles doesn't mean that those muscles need to be tight but just need to be strong for when you need to use them.For me, I think for keeping my leg underneath myself properly or changing my leg position independently of my upper body. Again, I am not sure because I was never taught proper equitation. I'm just kickin' an a pullin' ..LOL!
Another exercise I did was keeping by upper body in position and lifting my lower leg up so I could grab my spur with my hand. It also used those muscles and stretched the top of the thigh. Those two exercises may have been more about reminding me to keep my core engaged but I really felt it not only in my core but those muscles that ran along the side of the hip joint.
I DON'T LEAD 'EM AND FEED 'EM, I RIDE 'EM AND SLIDE 'EM.