I used to get grief from more experienced horsepeople 60 years ago because I would OCCASIONALLY let my horse graze some during our longer rides.
We’d be going on for a while, passing lots of grass and yummy trees and bushes, then we would stop and rest. Often I would stop my horse by a luscious, clean looking patch of grass and tell him “OK, time to eat.” When I wanted him to stop it was “ENOUGH”, enforced by a mild yank on the lead rope or rein if necessary.
Sometimes my horses would act like they felt not-so-well, then I’d put the halter and lead rope on, lead him into the woods or find an overgrown weed patch, and tell him “OK, time to eat” and he would zero in on ONE species of vegetation. Usually in this case I’d let him eat it as long as he wanted to, usually just a few minutes. When he stopped eating it I would tell him “time to go” and lead him away.
He did not abuse this privilege.
However I had a Paso Fino mare that would try to abuse this privilege, and guess what, I did not “let” her eat during our rides. Occasionally we had a few arguments about this but I was consistent and she learned that I would NOT just let her grab a snack whenever she wanted one. The first horse was my angel from Heaven, the second horse could be a mare from Hell when she felt like it.
Vladimir Littauer’s training system is based on teaching voice commands on the lunge line during initial training, and using them pretty frequently at appropriate occasions during riding. ALL my horses learned voice commands, from foal to fully mature horses. “Common Sense Horsemanship” and “Schooling Your Horse” by Vladimir Littauer have thorough discussions about training voice commands.