Current mods that can allow you to buy horses for your followers to use:
...this mod once installed lets your follower have and ride an horse that will spawn near your last ridden horse.
It works with stolen horses, quest horses, but of course works best with stabled horses.
1 - I suggest you to never mount the follower horse, it may result in the follower mounting yours and after you dismount you'd have to catch your horse that tries to go back to stables.
2 - You can fast travel but note that your follower may not be teleporting with you if you were mounted, IT IS NOT AN ISSUE tho your companion will come back after a while (specially if you transition to doors etc)
3 - Once their AI kicks in followers are rather slow (they walk) to catch their horse ( just go.. they will catch up ).
4 - The AI will not enter combat unless you dismount and so the normal behavior starts.
5 - If your follower is very close to you and you dismount, they may not dismount at first... but once you start moving they will.
6 - Follower horses are not owned by the player so they will show with Red name.. this is totally normal! they are not supposed to be mounted by you!
7 - If for some reason follower horses aren't showing up, mount up on your horse and then dismount.. in few seconds follower horses should show up.
This mod makes your follower ride a horse if (s)he has access to one he owns. If he has one near he'll mount when you do and dismount when you do. Sometimes it takes a little while for him to decide to mount, especially if it's the first time.
Horses for Followers will make new horses available to buy at all the main stables Better Follower Improvements will enable your companion to ride them and follow you. To make followers mount their steeds, all you need to do is mount your own and they will follow suit. When combat ensues, they will dismount and engage in combat as normal. The horses are set to avoid combat if possible, unlike the vanilla players horse. Now that you have two horses to pay for, you will appreciate their increased survivability.
The horses will be owned by the current Follower Faction or Player Faction, which means that even if you change followers, there is no need to buy a new horse for each one. Hope you enjoy and let me know of any bugs or mod conflicts. If a follower refuses to mount, use the Bill of Sale in the inventory (Books) to switch to ownership more suitable for your companion at any given time.
The HorseRidingFollowers mod description also states that you can make your follower ride non-horse mounts (like Mammoths and Wolfs) with this mod (and also, probably other similar mods): Like a Boss