Here's an interesting way to experiment with how your body affects your horse as you ride.
Ask your horse to walk forward. Establish a nice free moving, rhythmic walk, and allow your body to move with the motion of the horse. At the walk your seat will be following a more or less backwards and forward motion, your hands will be following the horse's slightly bobbing head.
Now stiffen your body. Stop moving with your horse and make your whole body; neck, torso, arms, hands and legs, rigid. What happens? Very likely your horse will slow down in response. Your horse's rhythmic forward motion will be become more stilted. Start moving in sync with your horse again, and you'll resume that nice free and forward walk. You'll probably both be more comfortable as well.
Now try stiffening only one part of your body. What happens? Is it possible to be stiff in one part of your body, while being relaxed and supple in the rest of your body? How does your horse respond?
What this experiment may do is help you discover any stiffness you may have as you are riding and how your body can influence your horse's gaits. For instance, I often let my neck and shoulders get stiff and hunched. When I relax my horse becomes more responsive. When I tense up, it makes it very difficult for my horse to be supple, as she carries a 'plank' that is working against her. Even something as small as tensing your jaw can be picked up by your horse and result in resistance and stiffness! Consider how a rider's stiffness might contribute to the horse's comfort and responsiveness, whether you are riding in the ring or on the trail.