The way I think of this is that, I can ask physical questions about a space-time which are impossible to answer unless one knows the full space-time, and hence I am inclined to believe that gravity is non-local.
For example, sitting at a point in a black-hole space-time I can ask if the given point is inside or outside the black-hole. As far as I can see, there is no local measurement/calculation that can be done around that point which will give the answer.
One has to know the full space-time Penrose diagram to know what is the complement of the past of the future null-infinity to know if such a region (black-hole) exists and then ask if the given event is inside that region or not.
The very definition of a black-hole itself looks non-local to me! I have to know the entire Penrose diagram to know if there is a black-hole or not!
What are the most precise ways known to justify that gravity is non-local? How does one reconcile this with the fact that Einstein's equations local? (any differential equation is local by definition!)
How does string theory and/or AdS/CFT see this non-locality?
Similarly isn't the existence of space-times with closed timelike curves an evidence that gravity is also non-causal?
What are the most precise definitions of "locality" and "causality"?
It would be great if people can refer to best known papers on these issues and if they connect to any currently exciting research question.