I do let my guy graze on the trail, but only after I have asked for a stop. And he stops grazing and starts moving with just a kiss or a light squeeze of the legs, just as I'd expect if he was standing quietly and not grazing. And if he were to ignore that, he'd get a bump with the legs followed by a hard smack if he ignored that, again, just as he would get if he was ignoring my cue to move forward at any other time.
I fail to see the difference- if you ask them to move and they ignore you, the consequences should always be the same, no matter what they're doing. If you're not consistent, you're more likely to confuse the horse and open yourself to having to deal with all sorts of bad habits. If this is a regular problem you're having, that means you are not training your horse to do anything other than ignore you, since if what you were doing worked, the problem would already be solved. Be assertive, tell her how it's going to be, enforce it a couple of times, and you'll likely have the problem fixed save for an occasional reminder depending on how often your girl feels like she can test you.