My greys would do the same thing if a cat were in our yard.
Two of our neighbors were fighting because one neighbors cat was in the others back yard and the dog killed it. Animal control said it was the cat owners fault for not keep the cat on their own property.
"He knew he was a bad dog when he ate the pussy" Of course they have made t-shirts that said that. I believe you could get them in O.C. Hubby had a couple. Hilarious.