First, I find that STRETCHING (esp by back) on a regular basis and before riding really loosens my hip and back. Yes, you need core strength, but you also have to be loose enough to follow the horse. Second, a trot that is well connected and over the back is so much easier to sit than a disconnected one.
I make sure that I’m sitting “on” my seatbones and not leaning back or forward. I think of having erect posture in the saddle (not rounding my lower back). I let my hips follow the movement without rounding my lower back. For example, during the trot, my hips move a bit like I was running - right hip forward, then back, the forward…-in motion with the forward and back motion of the horse’s hind legs. I control the stride length by how much I let my hips move with the motion or drive the motion - that involves the core muscles. In the canter, my hips are more stable and dictate the lead, but I let them move back and forth (forward and back and slightly up and down with the phase of the canter) to dictate the stride. I momentarily stop following the motion, which my horse understands as an ask to sit and collect a bit, for a half halt. But just for a moment each stride as I half-halt. I exaggerate the movement to ask for a medium/extended gait. I turn my hips slightly to signal haunches in or out/half pass.
Additionally, I “scoop” (??) alot with my seat and thighs to tell the horse that more pressure from the outside leg or inside leg (in pulses with the gait) means move your specific quarters laterally, not a more forward or upward gait change. So to dictate stride, my hips move but on roughly the same plane (say x and y plane) but to do lateral work, my hips also incorporate up/down, or the z plane of movement. So to shoulder-in, my inside hip sort of is lower at a point in the stride than my outside hip to say (in conjunction with my rotated shoulders and bending aids) “please bend, but cross that front leg and lets face it, slightly the inside hind, to stay bent but moving on this straight line”. I feel the shoulder-in comes mostly from my positioning to pre-start and then all from the motion of my hips and the outside rein aids to keep the bend on a straight line. I think the shoulder-in to haunches-in comes primarily from the hips (from that scooping movement in the z plane from the inside hip to the outside hip).
From the waist up, I don’t move much. I turn my shoulders to influence the front legs for lateral work.
I don’t know if this makes any sense, but this is how it feels to me!! Hips/pelvis and thighs and calves and heels work independently but in an orchestrated manner to achieve the proper throughness to sit well and small changes in your weight and pressure application/distribution is the key to providing effective aids and staying in real balance with your horse. It is SOOOOO much easier to sit a forward moving and through horse than one that isn’t!!!