Yeah, I would suggest pulling their head up or making them circle when they try to eat. If you circle them, then just do a couple of circles then walk on like nothing happened. Welcome to the forum by the way and if those are your 2 in your avatar, they are gorgeous.
When you want the horse to come along, kiss to him once and then lightly flick him with your rope or a training stick you have with you. You can flick him on the gaskin, under the flank, under the belly, etc. but do it lightly, but effectively. Soon all you'll have to do is kiss and his head pops up. Pulling on the head is like a fight really, and jerking is just rude, so this is a really easy way to fix the problem.
I usually pull their heads up. Not jerk, but firmly pull.
I let them graze when we're standing still, though. Or if I take them out to the yard to graze. They just know when it's okay to eat and when I want them to walk.
Thank you for the responses, Everyone! I really appreciate it.
Yes, the two in my avatar are my AQH geldings who are coming home to live on my small horse farm this coming Saturday! I'm excited, nervous and thrilled but I'm trying to get myself together because the poor boys will be excited and nervous enough for all of us! They have been pasture buddies for five years and are very close so that is a plus.
I appreciate all the replies and will try what was suggested. Thank you!
I pull my horses heads up with a firm 'No!' or 'Nu uh!'. I will also walk them out into the grass and MAKE them stand, then only when I say 'okay' are they aloud to graze. Seems kinda mean, but it has worked well for me. They know they get there grass eventually, they just gotta be a little patient :)