My horses don't get to eat with a bit in their mouth....period. It doesn't matter if I'm leading them, riding them, or they are standing ground tied, they aren't allowed to eat.
I need them focused on the job at hand, not scanning the ground for the next available bite.
I have a horse that has been a bear to keep from eating under saddle from day 1. I don't know if it's just a strange personality quirk that is singular to him or if it might be because he's a mustang...I just don't know. The one thing that I do know is that if I allow him even one bite while I'm riding without correcting him, he turns into a monster about eating with every step...to the point that he completely stops paying attention to everything around him, including where his own feet are at. He nearly went down on me once because we were going down a steep hill and he dropped his head to get a bite and tripped.
Now, because it's the only thing I've found that even remotely works on him, whenever he even thinks about taking a bite while I'm riding him, he gets a spur to the ribs. He's gotten much better now that he only tries a few times a year, but I know that if I allowed it even once, I'd have the monster back again.
When my horses are saddled, this is the only time they are allowed to "graze"; with the bridle removed.