I used to ride with a Standard Poodle and he always stayed right with me. But I did do a lot of obedience work from the ground first. WHen I first got him, he used to “surf” by running behind my horse holding onto his tail . . . but once he got over that he was great. He’s the only dog I’ve ever had who could (and liked) to jump cross country fences with me. Sometimes he’d go first and then turn around to see if we’d follow!
I ride on public trails all the time. I always slow to a walk when approaching or passing a pedestrian. Many of the people who I see are scared of horses and most of the people who have off-leash dogs like to have a chance to call them in. Only once have I had a dog try to chase my horse. It was when I encountered a dog walker who had maybe 10 dogs milling around. Luckily, I was on a horse that has no fear of dogs so I merely turned to face the dog and “drove” it back to the walker.