MY DOG WANTS TO WORK THE HORSES?
Early on, our pups/dogs learn that working horses is not an option, ever. Horses are prey animals and will kill a dog with one strike. When they kick, they kick to kill in a prey/predator situation. The other reason we do not let our dogs work horses is because we work stock horseback.
It is important to teach a dog to be neutral around horses. They need to be corrected if they eye them up or try tracking them. Some dogs will varying degree of eye/desire to work will struggle more. Some dogs can be neutral in the horse pen while others need to be 20 feet away. If you want to work horseback, you need to get them to understand horses are not an option to herd.
HOW DO I PROGRESS TO WORKING HORSEBACK?
I use the same fundamentals when working with a pup on foot as I do horseback. If you need, and your horse is rope broke you can do the leash work horse back as well. You have a different degree of pressure. It is important for a dog to learn that the expectations are the same on foot or horseback.
I have had to carry rocks with myself with client dogs that want to work the horses instead of the cattle. I would correct the dog, keep the horse faced up as he wanted to heel the horses, and wait for him to look towards the cattle. I would encourage him to work as soon as he looked to the stock. If he attempted to heel my horse, I would chuck some rocks just to make it undesirable for him to do so.
Good stockmanship should still be the goal when working horseback. Just because you have the ability to outrun the stock, does not mean you should speed everything up!