Horses should be able to carry their heads in different ways during a ride. Both are true regarding your question on keeping their heads low. Yes, horses that stretch their head and neck low are stretching through their back (topline) and are often moving in a relaxed manner. You will see horses break into a trot and stretch out their neck on the longe line sometimes after they’ve been cantering and bucking for a few minutes. Yes, horses have to put their head down in order to transfer the weight to be able to buck.