Mass changes spacetime curvature, and spacetime curvature pushes masses around, back and forth in a grand dance!
Space-time is distorted by energy, rather than just mass, which reduces the number of things the universe has to be prescient to. We can further eliminate some more prescience, by thinking in terms of density rather than mass: The laws of physics stated locally require only (say) a number and a field, rather than a pesky integral.
"Space tells matter how to move, Matter tells space how to curve"
And asking these questions is a good thing. I've been sitting down and really thinking about special relativity recently, it's fun going through old papers and seeing about how to derive the algebra in the most smugly experiment-less way.
One of the philosophically more pleasing things about GR is that it is local. But, of course, Newton's conception is a small-mass / low-velocity limit, so how can that be?
GR says that the effect of stress/energy at a place x changes the metric at that place. But the metric is something made of derivatives, so the space in some small neighborhood (this is the local part) nearby gets deformed. That deformation is itself a form of stress, and so places in the neighborhood of x effect places THEIR neighborhoods and so on.
So there's nothing built-in that's long-distance. Big long-distance effects are built up out of everybody talking to their immediate neighbors.