There are two versions or stories to it. The horse was found dead near to Last Stand Hill which equates to various tribal history of Custer being 'knocked' off his horse and possibly by a Sioux or Cheyenne woman. There are a number of candidates for the role. It is also given that Grey Earth Track or Sounds the Ground, took possession of the horse and went with Sitting Bull to Cypress Hills north of the border. The Canadian authorities recovered a Custer horse which I believe was returned to the US military through diplomatic channels but this was a gray animal. Army mounts did not fare well on the Plains in being used to forage (oats) rather than nibbling only grass. Of course one of the horses which Custer owned, survived and was sent to his father True West link
I though I read somewhere that two officers found what they believed to be Custer's horse, cut off the hooves and kept a set each. One of those officers lost his set, while the other had his set made into ash trays. Don't remember where I read that, or how they though it was Custer's horse, or how long after the battle for that matter.