The circling or backing up is very good suggestions. I think they are both so effective because they are moving, but getting no where. When I was working with my horse just at halter a year or so ago, the circling helped tremendously with keeping her in check with me. Every time she'd start to walk out ahead of me, we did a circle and tried it again. (Oh, & btw... these are small circles we are talking about.. not big rounded out ones. Lol) After not to long of this, she realised that if she didn't stay with me, she'd still have to do a lot of movement, without any progression in the actual distance traveled. (With consideration of how horses are in the wild, you can see how this is something the horse will usually try to prevent from happening. In my horses case, stay with me, we get somewhere. Don't, and we're staying here for a while.) Sorry.. I don't really know how else to explain it. Lol.
As for the grazing on the trails, definitely one of those things you don't want to become a habbit. My uncle has this horse who he's had it's entire life (now 20something), and has always let it graze while riding. He let his kid ride the horse that day, and my other uncle (who was riding with them at the time), said that the horse just flat out stopped on this one hillside (that mind you is very steep and rocky) and started munching on a patch of grass sticking out of the rocks. Definitely not a 'wonderful' time to eat! But, being that this horse always got away with it... did it anyways.
Ok, now for my own personal experience with the issue. Rofl. My horse Misty (who is still green, 6 years old, and had last year off for having a baby.. we bought her pregnant the fall before and didn't know lol) got to be ridden again this year, and that was probably the biggest problem I ran in to with her on the trail. (She was GREAT otherwise... didn't have any other problems with the unbelievably huge amount of down trees from the bad winter before, water crossings, porcupines, etc. lol) I didn't have as much problems with it on our half day rides, because being green she was more interested in all the things around her then her belly. But at the end of the summer we went on an all day ride in the mountains, and all that grass looked pretty tasty to her. Though tedious, I kept on her about not eating, and though reluctant at first, she soon gave in. It went from a quick 'bite the grass as we walk by', to 'put head in that direction and if no threat from mom take a bite', to 'fine, I'll stop bugging you about it'. Horses will test you and will be persistant about it, but if you stay consistant with what you're telling them, after a while they usually give in.
Anyways.. yea. Lol. Sorry... I'm just a firm believer in telling personal experiences as answers, rather than 'do this'. Which tends to make my posts long... but, I think that you learn and can make your own decisions better when you know what others have done in similar situations.
Hope this helps!