How annoying. I know it may not be your barn, but here are some ideas.
1)Sweep up all spilled grain as soon as possible.
2)Introduce predators like cats; remove any food every night.
3)Put up a plywood roof structure over your water trough that your horses can
drink clean water.
4)Set traps in the favorite pigeon roosts, and check them often (the most
immediately effective, but also the most time-consuming and involves
ladders often). Barbed and sharp edges on the roosts are somewhat effective.
Sticky tacky pest stuff works, but it's messy.
5)Encourage barn owls to come (provide owl nesting shelves in barn corners
not above water troughs). Contact local Audubon Society and Raptor rehab
centers to let them know that you want your barn used as a release/relocation
spot.
6) Make sure the gun-owning neighbors know that you don't want them to
shoot the native hawks & owls; if they object let them know about the pigeon
trouble in your barn. Some people shoot them because of a few free-range
chickens and ducks taken. Large wire-covered litter pens and duck ponds
are the ages-old answer to that. Then, if they still lose poultry, the problem
is some other varmint.