Goats will eat mesquite voluntarily. In a rich environment they will pick around, and not necessarily eat the mesquite first, but they do like the leaves. (I believe cows will eat the seed pods too).
I don't know how sustainable a solution this is though. If you ever see goats in a patch of mesquite, all of the trees are happy and leafy up top and just pruned clear up to about 5' high where the goats are eating. Most other vegetation and grasses are choked out or not getting enough sun because of the thick mesquite.
Perhaps if you regularly cut down the mesquite, there would always be new growth down low for the goats to munch on.