They had single shot carbine, Springfield, but long-range carbine, which was better than short-range repeating carbine used by Indians (to shoot while being mounted)
They did not. They didn’t even have real saddles that are made for such a position.
They did tie loops in the horse's mane to pass an arm through so as to hang down and shoot from beneath the neck of the horse and they were pretty damn nimble when a horse was shot out from under them.
The best chance to kill them was to wait for the horse to go down and shoot him while he is running for another horse.
The Sioux typically made such charges in the middle of a herd of all of their horses, so it was very, very tough to kill them, especially when a thousand or so horses are running.
That Custer, also known as "cinnamon," to some of his contemporaries, made such a breastworks while waiting for the rest of his commanders to follow orders and in the middle of battle, and to then hold out there until the last man was killed in the final charge is a testament to the bravery under fire of him and his men. - Reno should have been hanged.
Horses were power and wealth to the tribes. Depriving them of them quickly settled things, or as a cavalryman of that era would say; “without a horse a mans a foot”
- Thanks to great granddad's memoirs.