Hi GC! I don't claim to be an equestrian expert either but I did do mounted ACW cavalry for 12 years as a reenactor, so I do have a little experience as well as having studied the all the most commonly used period manuals out of necessity. There was simply no drill where the horse holders dismounted.
Verbal Order: "Prepare to fight - ON FOOT"
..Dismounted Troopers: The dismounted troopers hook up the saber, unsling the carbine, and move to form double ranks 12 paces in front of the linked horses, and come to the position of Carry - ARMS. (The commander may fight dismounted in a double rank formation, or he may advance the dismounted men as skirmishers.) Horse Holders: The commander places a junior officer or NCO in charge of the horse holders, who remain mounted and have control of the linked horses.”
There an absolutely great source for ACW cavalry drill here (I've worked with these guys a lot back before I retired) https://www.secondcavalry.net/drill
'Son of the Morning Star' is one of my all time favorite movies, so realistic for the most part you can almost smell the saddle soap! There is that scene where some warriors try to scare the led horses and the trooper in charge of them dismounts and he's suddenly not a hose holder any more as all four gallop away.....The army doesn't like this happening over much. :face_with_rolling_eyes: But as a rule I would caution against using movie and TV scenes as reference material because those scenes are shot using professional wranglers in uniforms supplied by the wardrobe dept.