I did a similar horse/foot race in Montana. The format was gauging your horse's endurance/speed and rest. Both the horse and the man had to cover the entire race distance, you just chose when you hopped on and off your horse to give your horse some rest. That race was over 18 miles, and I ran 12 of them while the horse trotted next to me and my team won by a considerable margin. If I recall, all of the top 3 teams, the runner covered more distance off the horse than on. Now, this race was part of a Native American Pow Wow, and there were some really strong native runners, but most of the horse were not particularly trained to cover distance with any sort of speed. My horse was a roan appaloosa that was primarily used for ranch work and a little bit of trail ridding, but was quite playful and athletic.
From that race, I can offer a few observations:
1. I would have been faster running over 18, than riding the horse over 18, but the horse was faster up some of the climbs than I would have been by myself. The horse just seemed to tire fairly quickly when I was ridding it, but seemed to stay with me with little trouble if I was off it. (I did try to "gain time" on some flats by ridding, but that strategy didn't work). My horse didn't have any specific training for moving at a steady clip over that kind of distance though.
2. Most horses really struggled over the technical stuff that my horse seemed to be faster on. If the trail was a bit more defined and well-traveled, I think all the horses would have been faster over climbs.
3. As the race went on, the horses notably tired. I wouldn't have even ran my horse in retrospect. I thought a few pack/hunting trips of 30+ miles would give me confidence that she'd fair well, but she was exhausted for the last 4 miles and required quite a few stops towards the end.
I think horses can be trained to be quite quick, but if it's just an average horse vs. a fairly well-trained marathoner (sub-2:30), the runner wins every time. If the horse is calloused from fox hunting or just long hard running, it seems like they should be quite quick.
A lot of people think that runners would win over any animal as the distance gets long enough, and that's true to an extent. I don't think anyone would run a horse over 50 miles at a clip that a runner can cover that distance...but I don't think any human would beat a sled dog over any distance regardless of terrain.