The first argument would be equivalent to saying that each time you moved, the Sun would disappear until you came to a stop and remained stationary for 8 minutes. In reality, if you move, you run into light that left the Sun even before you started moving, so you don't see any difference unless you happen to be moving incredibly fast (you'd see a shift in the frequency of the light you're running into).
Just because the distance is part of the equation for determining the force of gravity, doesn't mean that only the 'gravitrons' (or whatever) originating at that precise instant determine the gravitational attraction (the equation isn't the reality - it describes the reality). I would think any gravitational forces an object generates exist regardless of whether there is any other body there to interact with. If another body does exist and is in motion, it, too, has its own gravitational forces that are moving with it. The equation for determining the force of gravity describes the interaction occurring where the forces from the two bodies intersect each other and is independent of the speed either body is traveling - at least, unless you're traveling some very high speeds, at which point I'm not sure what kind of difference you'd perceive that would be equivalent to the frequency shift of light.
Just out of curiosity, what kind of difference do LIGO and the Gravity Probe B expect to detect?