Griffins become an apex predator they would have no other creatures that could harm them ( except maybe the large moose). The additional ability to fly would not only make a ambushing prey a lot easier. Chasing down and overpowering their prey would also become easier. The usual tricks that prey animals use to get away from a mountain lion wouldn't work on a Griffin.
Because of this the population of other animals be completely devastated the Griffins hunt them to extinction. This would eventually backfire as soon the one of the any food the Griffins to eat. The population of Griffins would drop quickly. Executions population go down the population of prey animals would eventually. The rise in available food causes the population of Griffin to rise again. This would repeat itself over and over again until you would eventually reach an equilibrium.
In the end to summarize you would have a world very much like today except for with Griffins as the apex predator ( other than humans) with some species of prey animals extinct.
Prey animals that didn't it go extinct, would probably do most of their grazing near cave or maybe a very thickly wooded forest or some other place where they could cancel fly advantage of a Griffin. If they somehow managed to detect the Griffin before he ambush them they a retreat into the cave.