Some of the large chain grocery stores won't give away or even sell the bad produce/bread that they are going to throw away, the corporate policy is that it MUST be destroyed and not even employees can take it. I've run across that issue before when trying to get scraps to feed my goats.
My college roommate worked at a non-grocery that had this same policy, and when we moved once, they would not even let us get cardboard boxes! They would throw them into an incinerator as soon as they were empty, so that no one could take them! Any damaged or unsalable merchandise would be smashed to uselessness and then thrown out so that no one could get it out of the trash and take it home. Part of my roommate's job was to make sure that anything that could not be sold was destroyed to the point that it could not be used. They would not even let her buy damaged items at a discounted price when she offered. I think that's pretty ridiculous!
It's a great idea though, if you can get safe to feed scraps for free or trade. The locally owned grocery stores here donate all their non-sellable produce and bread to a homeless shelter, so there's none available to get for my livestock. I'm still trying to convince my chickens that kitchen scraps really are food...I think they did not get anything to eat but scratch feed before I got them, so that is all they want.
The biggest issue with the grass clippings, aside from whether the lawn was sprayed with pesticides and fertilizer, is not with the moving process. The tractors used to mow hay also run on gas just like a mower, and that is obviously safe. The issue with animals like horses or goats is that if they are not used to grazing on green grass, the clippings are very rich and overeating on it can make them very sick. Also, there might be bits of trash or metal/wire in there that the animals could ingest. I've fed lawn clippings to all sorts of animals before, and it works ok if you just be careful. Check it over for trash, and don't let them overeat fresh green grass. If you're looking for something like that to entertain them, you might also ask feed stores that sell hay if they would give you some of the hay scraps that build up on the ground around the bales. My chickens love sorting through my goats' hay leftovers.