I also have this type of issue with the ocupier of land ajoining mine he has around 20 horses and a dozen dogs loose on a few acres of scrub, (he lives 5 miles away) because they have little to eat the horses are constantly knocking down the fence/hedge to get in to my fields, & with the wet ground over the winter they have done a lot of damage to the ground, around 6 of the dogs are left to run free all the time & are constantly on mine.
I have had the RSPCA out they are a waste of effort, the local dog warden tried to help but has no power on private land, and unless/untill a dog "goes for you" you cannot shot it even on your own land, and the horses are classed as "livestock" & while he is responsable for keeping them in it would aparantly take legal action to enforce it.
Re Deer or Pheasants, if on your land you can shoot them as per any legal seasonal or calibre conditions.