Since propagates at c and all of the massive objects associated with gravitational fields move at classical velocities, the affect of the motion of an object can have very little affect on the fringes of the field... BTW I don't number crunch anymore. I do tend to speculate about how different approaches to theoretical physics intermingle, so to speak. Really keep in mind that I do SPECULATE on issues like this a lot. I don't know with any certainty any absolute answer. So in the speculation mode, once you accept GR as the basis of cosmological dynamics, space and spacetime (depending on your interpretation) become dynamic components, not just the massive objects, sitting at the center of gravitational fields. The question then becomes, does the gravitational field described by spacetime, move with the massive object at its center? More directly I think, to your question.., the gravitational field of the sun that the earth experiences right now is some 8 minutes delayed. It does not get stronger and weaker depending on the time of year and where the is earth in its orbit, relative to the sun. Gravity seems to act toward the instantaneous locations of the objects involved, at least with respect to orbits in our solar system. That suggests that the field does move with the source, (specualtion again) perhaps an artifact of the billions of years of linear frame-dragging, the motions of the involved objects have had on the involved space/spacetime. Which again must be thought of as dynamic components of the larger inertial system, we call a solar system or even Galaxy.
