Depends on how long the distance is. On short distances over only a few hours, fast quadrupeds are all much faster than people. Over a full day or several days, people are much better at dealing with exhaustion and can keep up a pace for durations that animals just can't.

Though I found out that there are endurance horse breeds that can make 160 km in a day, while human athletes can make 120 km. Don't know if these horses could pull of the same feat again the next day, though.

Camels crossing the deserts in central Asia can do really long trips in one go, but apparently a good number of animals died on each crossing and after they made it to the other side they had to recover for months before being able to make the return trip.

But that's horses and camels, which evolved for open steppes. Deer are native to forests and to my knowledge don't do long migrations (reindeer being the exception), so they should be doing much worse on multi-day endurance races.

In practice it doesn't matter, of course. I use the same overland speed table for all characters and animals, with heors simply having a relatively low Strength and not being able to carry much before losing speed.