Move with the horse is good that means you are letting the movement through. Start sitting deep without stirrups. A dressage saddle will teach you because if you don't sit properly you feel really u comfortable, if you sit good is a pleasure. Start backing yourself over vertical in transitions and hold ypur body tight on to lower paces transitions. No stirrups. Just canter the few steps you can hold on property. And increase that steps with time. Better a good straight line than 2 messy rounds in canter. Keep your knees and calls squeezing the horse to balance yourself.
Hey Emeline, this is something that took me forever to learn. You have to learn how to move your hips with the horse and to keep your upper body and legs independent from your hips. It takes time so don’t worry if you can’t get it for a few weeks or so, but I would suggest taking your stirrups away and work on feeling your horse and how he goes. Also focus your attention on sitting up straight and proud, even a little bit back with your shoulders back. Keeping your shoulders back and your leg long and down will help find the correct canter position the easiest, keeping a long leg around your horse will help keep you secure in the tack :) Hope this helps!!
Take your stirrups away, the only way to sit is deep in the saddle and not tilted forwards otherwise you will slide out the saddle, it sounds scary but it’s amazing how the muscles in our body work to secure our positions to stay on the horse, only do a few minutes to begin with each day so the muscles don’t get fatigued, if they get fatigued then they won’t be so effective. Gradually build up to increase strength over time. Do not rely on your hands otherwise that puts unnecessary pressure on the horse’s mouth
I sometimes experience the same problem. One of the first things is having strong legs. The part of your leg in between your heel and knee should be touching and and applaying strength to the saddle. An other thing is a good position in the transition to canter. Also you can practice light canter (or show canter) and then sit back ok the saddle for a while. Repeat that position variation till you are more comfortable. The position in any transition is also going to develop the rider's position.
Hope your question was answered.