D&D 5E (2014) Miles per day on horseback: how many?

Just to clarify, I don't recommend (or do myself) for DMs to ALWAYS mess with horse (or mules). I do recommend that DMs be aware of some of the ways in which they can use mounts (or hirelings/retainers/etc) to spice up of boring parts of the story or even add interesting parts to the plot.

It's just another tool for the DM. As with any tool, it should be used for fun, however the table defines that.

Fair enough. Most y’all like that I hear is from DMs that delight in doing stuff like that frequently, brag about their players not trusting anyone or anything, and still somehow have players.
 

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Are they ever in a hurry or have a lot of stuff to carry?

Sometimes they are in a hurry, but mounts haven't seemed to be relevant yet for those situations. Urban, jungle with raft, that sort of thing. They've rarely had problems with their carrying capacity, and since they've been to Sigil they now have almost enough bags of holding to go around the whole party. Of course the latter mostly is about pack animals, which won't speed you up, but I do like the idea of occasional pack animals coming along. There's just something interesting to me to bringing along beasts of burden or transportation that apparently my group doesn't share.
 

Here are the results from a basic Google search, with options for Phantom Steed added:

A horse's traveling speed over distance varies by gait, but a comfortable, sustained average is around 4-5 miles per hour (mph) at a walk, while a trot can cover 8-12 mph for extended periods, and a canter tops out at 15 mph. While horses can gallop much faster (25-30 mph), this is not sustainable for long distances.

Consider there are many factors when calculating the distance an individual horse could cover, including the horse's fitness and breed; the terrain; regular breaks and downtime/rest; and of course, the load the horse is carrying. A horse carrying a rider will travel a shorter distance than a regular horse, and let's face it, no one here cares how far the horse can go without its rider.

So, considering a horse's movement speed of 60 ft. in D&D, this means that a horse walking steadily for 8 hours can cover roughly 32-40 miles. Assuming the party has the common sense to allow 3 or 4 short rests in there, this can also be equated to a 12 hour day of traveling assuming normal breaks for food, water, and rest for both horses and riders.

The DM may allow you to modify this somewhat by incorporating trotting and cantering, with regular walk breaks, a horse could potentially cover 40-50 miles in a day (though this is not advised in difficult or unfamiliar terrain, as a cantering horse can easily hurt itself - gopher holes, ditches, bogs, etc.).

Galloping horses can cover much greater ground over very short distances, but require a rest twice as long as the time galloping, so it doesn't save time over the course of a day, but could come in very handy when you need to leave a specific area very quickly for a hour or so.

Phantom Steed: Now... here's the fun part. Phantom Steed creates a horse-like mount out of nowhere that can move at 100 ft. speed. This is 60% faster than the Riding Horse stat block's speed of 60 ft. True, the Steed only lasts for an hour, but then again, it doesn't require food, water, or rest like a horse does... it only requires 11 minutes after each hour to cast it again as a ritual. So, you could be fully justified as a DM (or as player arguing with your DM) allowing a PC riding a Phantom Steed to extend their travel time and distance by an additional 60% ... maybe even 70-75% since they don't need as much rest as a live animal.

So, by this logic, a horse might get you as far as 40 miles over a given day (20% faster than the 32 miles done by a walking human). But a Phantom Steed could bear its rider/caster somewhere between 64 and 72 miles in a given day. On a grid map, it's just 100 ft. vs. 60 ft. speed. But over the course of a full day of travel, it's even more, with no repercussions of possibly endangering or injuring the mount. (This is why every wizard I have learned Phantom Steed as soon as possible. It's awesome!)
 

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