to a horse, being a follower is usually the most comfortable place to be. So, when we use the word "Dominate" , is comes with human inferrences of quashing the will of another, who like the dominator, wants to be free, and that isn't necessarily an apt way to describe a horse.
A horse wants to follow, to be guided. he is safer that way and can relax a lot more. But, for him to be able to follow, he must know that the leader is a leader. So, he may push out against the other horse/human, to find the boundarie of that being and see if it is strong enough. If it is not, then it will give way from his pressure , and the horse will find itself in a vacuum, and then HE must lead; with all the encumbant stresses and possible dangers of leadership.