I use a .220 Swift with either Barnes Burners VLC 50 grainers or Hornady V-Max 50 grainers over 38 grains of IMR 4064 in Winchester nickel plated cases. This does a number on the dogs, as a matter of fact day before yesterday I made a head shot on one with a V-Max, and (kinda graphic here) it dislodged the eyeballs from the sockets. These are very accurate loads holding about .2 inch groups off sandbags on a calm day.
By the way, I used to use a 7mm Mag, and that got a little too punishing to be shooting at dogs, and the muzzle blast spooked the crap out of the cattle, horses, calves, etc.