There are various Hellhound/Devil Dog legends in Britain. Specifically, the “grim” part of the name and the reference to them haunting graveyards suggests the Kirk Grim, which hangs around churchyards to protect the dead buried there from evil spirits or the devil.
There are many Devil Dog legends in Sussex, most of them on, yes, the Downs. Most of these creatures are described much as the grimhounds, and to see them is a portent of death: presumably if they’re visible to you, then you need their protection (and so are or will soon be dead).