Canids and equine are natural enemies, yet also natural playmates. A most interesting but I think perfectly normal situation.
Allow your dog Ziggy around your horses for awhile only when the horses are tied or otherwise restricted. The horses need to see him as something other than a predator to be fought off.
One of the ways to incite more energy in the game, Chase The Tiger, is to move the target lower to the ground past the front of the horse. That triggers the old fight instinct from "flight or fight." If you do it near the rear of the horse you have a similar effect, but more likely to evoke a flight response. I don't recommend it because should the horse choose the fight response it's going to result in a kick, something I don't want to encourage.
Let me make clear as well that while we describe such behaviors in terms that are human the animals probably experience somewhat differently according to circumstances.
I don't think our domestic dogs, if fed properly, view a horse as a meal, nor do I think the run of the mill horse dog encounter makes the horse think that the dog is out to get him and he must fight the dog off.
From the Mother MareTM perspective these behaviors, stalking and attacking, or running and fighting, are both built in and acquired by practice. Watching young horses at play shows clearly their tendency to buck and kick, feint attacks of biting and striking, as they practice being a horse with each other and with their adult caretakers, mom and aunties.
Puppies roll about with each other mock attacking, chasing, being predator, being prey too, with each other - from instinct and also to acquire a sharpening of their skills.
Both are playing. Play is the work of the child. How they become social and milieu management proficient.
Many, probably most, even all, retain a sense of play if the situation allows for it.
This is why with horse-dog encounters if they are accustomed to each other play is possible - not guaranteed, but possible. If you think about the encounters in this way, first familiarity with lowered risk, and then play dates the play portion can be developed so the triggers - the dog running past or at the horse, or vise versa, doesn't result in injury to either.
There are three times one wants to be careful of not getting kicked by a horse: when they kick out in play, when they kick out in fear aiming carefully, and when they kick out and you don't know the reason - oh, wait, that's all the possible instances. LOL
In other words horses kick and sometimes the best we can do is be alert and keep clear.
Personally I tend to do far more "taming," (desensitization) of the hind quarters than I do the forehand, though I'm perfectly aware that a strike can be as damaging and dangerous as a kick.
The dog has the same problem we do in avoiding the bites, strikes, and kicks of the horse.
As time passes, and it can take a long time, the dog will get the measure of the horse and know how to skirt outside the kick, bite, and strike ranges of the horse. Labs aren't very good at this but they can pick it up. Herding breeds of dogs are often extremely skilled at the game, able to play vigorously and drive the horse about, and also to let the tables turn and take up dodging and running and get the horse to chase them by staying just out of range but still close enough to trigger the horses' fight response.
In other words, playing Chase The Tiger with the horse with their own body as the target. They can become, as Romy pointed out, great friends and companions owning each other sometimes more intensely than a human to horse relationship.
My hunch is Ziggy will be doing some avoiding for awhile until he get's the measure of Corado.
Bonnie is sensitive to scoldings. She has chased and hurt Rio, and I've bawled her out for it, and she has gone off in kind of a huff, but greatly diminished her chasing behavior.
I think Romy most definitely has the solution. Supervised visits. Safety. Intervention when needed.
I become, over the years, more and more convinced that being congruent with the horse is the key. I express my feelings very directly to the horse - approval, or disapproval. It's very Mother Mare like, I think. And handled well and with less anxiety then being too clever and frightening the horse by inconsistency and incongruety. I do NOT smile at that horse when I'm frightened or angry with them. I growl and I yell and I crouch a bit too if I think that much language is needed.
But, like Mother Mare, I am quickly and completely forgiving when the dust settles.
Donald, Altea, and Bonnie Cupcake who has chased Rio many times without catching him very often.
Love is Trust, trust is All
So say Don, Altea, and Bonnie the Wonder Filly.