as far as i know, horses are animals, thus they get TIERED, need FOOD, REST, so they cant run for a WHOLE DAY!
and nor can a human, unless hes a friggin olympic athlete!
so i propose the following:
based on horse type horses get tiered, thus their health drops and they move slower.. and if it comes to like 10hp(out of some 100, since horses have mroe health than humans) the horse falls and needs to be cured or else it dies.. something like first aid is needed..
a horse can be run dead(literally) if you gallop on it for too much time... also horses carry a certain amout of weight(i think its already incorporated) so a wagon horse can carry some 400pounds but goes 5unit speed, a warhorse carries 200pounds, with 9unit speed, a racing-horse(courer?) carries some 150pounds and runs with 11unit speed... etc
as for humans, i very much doubt that they can outrun a party with 10horsemen, even though they move slower than normal horsemen(a party)..
for example, in medieval times from one castle to another is one day's hard ride for a messanger reporting a war and need help, else the british would still be on where france now is...
hope you can understand whatever i wrote