At that strategic point the rider stops, dismounts, ties the horse to a tree, and continues down the trail on foot. The team member who started on foot gets to the horse, unties it, mounts, and rides to catch his partner up ahead. When he gets to his running team member he can either stop and exchange, or ride further up the trail and tie the horse to a tree and then continue running. When, where, and how a team exchanges is up to them, and this is where the strategy lies. Every trail runner has their strengths and weaknesses, and the same is true for horses.
Trail runners find the sport exhilarating.
It has a mind of its' own. It thinks and reacts, but not necessarily how you think it should. One cannot be a mere passenger, nor can one treat a horse as though it were a machine. It is flesh, blood, and spirit. It is a herd animal, and you and your partner are the herd.