grand-dog
First you have children. Then it seems that these days the children all acquire fur babies, presumably as a practice run at parenting. Finally they have human babies and you have grandchildren and a grand-dog. It does make it easier to answer the questions that arise as you walk down the street with the dog on a lead. “Oh no, this isn’t my dog. This is my son’s (daughter’s) dog. I’m just looking after it until ….” You can simply respond “This is my grand-dog.” It is easier to play pass-the-parcel with a dog than it is with a cat which may explain the many instances of grand-dog and the few instances of grand-cat.