Here is what you do, if you see equestrian hikers:
1) First and foremost, accept that the trail is open to horses, so don’t get engaged in a civil-war with the people on the horses. You both have a right to be here, so stop the bullshitting, grow up and cooperate.
2) The “risk” with horses is that they can get spooked by an unknown hiker, as the hiker looks weird and dangerous with a backpack and trekking poles. The horse can panic and start to escape or jump, both presenting a large risk for the rider and the horse, especially in steep terrain.
3) Follow the instructions from the rider: where do I place myself, what do I do? Normally, you want to get off the trail on the downhill-side to allow the horse to pass as you don’t appear as scary from down there. However, follow the advice from the rider who knows the horse.
Even on a flat section, get advice from the rider as this communication will likely calm the horse and you don’t want to be hiding behind a bush.
Hiking with horses is a discipline in itself and I will not dispense advice on this, as I don’t know anything about it. I only know one thing for sure: That the common conception that it is “easier” to hike long-distance if you bring a horse is completely misunderstood. Bringing a horse on a long-distance hike is like bringing a child: one more living thing to take responsibility for that will give you immense joy, but not much assistance. Yes, you can sit on the horse for some sections, but you’ll have to deal with challenges such as: where can it eat? where can I find a stabile, large water supply? what if it gets injured? And very important: how do I get the horse across that huge 6 feet downed tree that is blocking the trail? Bringing a horse is an admirably challenge, not a luxury.