I think the precedent is Rylands v Fletcher. You are not allowed to do things on your land that can be a hazard to neighbours.In that case water from ponds flooded a neighbours land (If my memory serves me right) and compensation had to be paid. If pheasants are damaging crops you make a claim against the person who released them.
mmmm whatis the legal position is re releasing birds onto land where you only have the shooting rights on , i'm assuming your neighbour has the release pens etc on his land& the shoot just hold the shooting rights to your land ,