They help with any type of fly that comes from horse poop or rotting stuff.
I'm not sure where the green headed beggers you asked about come from. I checked my bug book and the 2 green head/body ones I found are the American horse fly and the green bottle fly.
Predators wouldn't help with the horse flys since they breed in wet areas laying eggs on plants that overhang water. The bottle fly breeds in poo so it would help with them.
I also keep fly traps up all year to help with the flys that the predators don't kill. The horse flys are tricky *******s though and seem to avoid the traps.
I remember before I started using them this time of year the horses faces would be plagued by flys. Now there is just an occasional one or two. Now if I could just get enough to do the whole county, I wouldn't have to buy fly spray for trail riding