This is the major problem between general relativity and a quantum theory of gravity. We do not have such a consistent theory yet, and it is hoped that when we develop it, we will be able to resolve questions such as this in a consistent manner. Of course, we know of many paradoxes in physics that even the knowledge of a full quantum theory do not help us with. Gravitational fields as 'classical' fields do not travel faster than light, but the problem is that gravity is not JUST another field in nature, it is space-time itself. I do not know how to think about gravity in both ways at once ( as a field and as dilating space).