My horse is respectful I dont care if he grabs a mouthful when walking thru tall grass. If he keeps his pace and does not pull on my hands, it’s fine.
I will let him graze by telling “OK”, then when it is time to move on, I tell him, “last bite”, after he gets one more mouthful, I hold the reins, and urge him forward, and say “let’s go”.
So, I do both. I trail ride w/an upper level dressage rider, who wont let her horse eat while she is on him. We hardly ride for more than 2 hours, so it isnt a big deal. When riding w/her, I dont let my horse eat, out of respect of her. We do a lot of fast rides, so really, we stay busy that there isnt much time for eating.
I do longer rides, either solo or w/other people, and I do let him either grab and go, or let him graze for awhile.
Usually, he is very good, but there are times he does test me. Last spring was really wet, so he wasnt out on grass for awhile. When I would ride him, the grass would be almost too much for him to resist. If he did pull the reins hard, he lost his grazing time for awhile. If he was good after I corrected him, i would just stop and let him graze for a few minutes as his reward.
So, I dont have a black and white rule, it just depends. I do NOT like when a horse pulls the reins or stops his pace to eat, that is just rude!! There is a happy medium where the horse and rider can decide on when eating is ok. Some people dont allow it at all.
Endurance people do want their horses to eat and drink. It helps w/gut sounds and hydration. In Endurance, you pace yourself so you allow time for your horses to relax and cool down. During those breaks, you want your horse to graze. Normally, it is not usually trotting the entire ride.