This actually does happen a lot. Please know that your cousin will need to seek the state and local laws in his own state because they differ.
Who owns the horse? That will be one of the first questions asked, which then will determine if this is considered abandonment, or not, or other. Even if your cousin doesn't own the horse he's responsible to take care of this domestic animal/horse, for right now; shelter, food, water, and he's responsible for any of it's actions. He needs to call animal control asap, and inform them of the situation, and go from there. The good news is he knows who dropped off the horse, and assuming it belongs too the ex, then the responsibility will be put back on her. The rest depends on the laws in his state, with a court of law there to handle. Good luck to your cousin, and his daughter, lady4u.