I read that they typically had three or four horses following them whenever they march so they can easily change--but wouldn't the horses, which are also riding with them albeit without a passenger--also be tired once the riders change their mounts?
There's a big difference between carrying a rider and following along unmounted; the steppe mares favored by the Mongols weigh in at about 600 lbs, so a 180 lb man with some basic provisions is going to be a third of their body weight. It's much more efficient to shift the weight between your string of horses; imagine going on a hike with four friends, but you need to carry a single 60 lb pack. Everyone's going to tire out over the day, but it'll be much faster for the person carrying the load, so it makes sense to share it among the group.
Not necessarily. For example, imagine you are in a field running. Your body is designed to carry your flesh over your bone, and not much else. The same goes for a horse, and they can run for hours on end, but when you stick 150 pound lump of anger in human form on top, plus 50 pounds of laminar in addition to whatever marching kit and such an ostensibly nomadic fighting force will be carrying on their person, it naturally cuts down on how long they can endure a full out sprint, while the harem of horses they may be leading behind them will be a few degree's more rested.