Years ago I went and auditted a Charles DeKunfy (spell?) clinic in my area. I had only a year or so of dressage lessons, so I hardly understood a thing that he said. But one thing I remember him saying, something I have heard NOWHERE else , for some reason, was to use your upper body in rythm with your horse's walk to allow his forward movement it's fullest potential.
It's like this:
When the horse walks your lower body must belong to the horse's motion. There is a forward movement and a lateral movement and your hips must go king of rolling up and forward with each time the horse steps under himself. The right rear steps under himself, your right hip raises and movew forward with this motion. And vice versa for the left side.
Most riders will either keep the upper body perfectly still, almost a frozen block under which the lower body undulates. OR, they will actually counter the lower half of their body's movement by rotating the upper half in a countering motion. So, when right hip moves forward, the rider swings the left shoulder forward and vice versa. This really dampens the horse's forward energy and is a terrible waste of both rider and horse;s energy . NOT good !
But another choice exists, to amplify the horses walk;
When the right hip advances (the rider's) as the horse steps under himself, the rider can also, ever so slightly, advance the shoulder ON THE SAME SIDE. So, both rider hip and rider shoulder (to a lesser degree) advance as that side of the horse is also advancing.
It is hard to get the rythm and keep it as it's a bit counter intuitive. It is more natural for us to counter the upper and lower body to each other, like a pole that the lower half twirls one way while the upper counterbalances it by twirling the opposite. This feels natural, but like I said, it dampens the FORWARD movement of the horse.
Try it sometime. Go out walking and kind of listen to your hips and focus on one to start with. As it rolls forward, move the same side shoulder WITH it and see if after a bit you don't find your horse moving along just a fraaction freer.