The "force" of gravity does not travel at any speed, it is a field that exists everywhere in space. Changes to this field (i.e. gravity waves) travel at the speed of light however.
BTW the example you gave of moving an object back and forth and detecting the changes in gravity is exactly what a gravity wave detector does. But the objects that they hope to detect this way are things like merging black holes, which generate a lot of gravity waves, theoretically.
A long while ago i asked if gravitational waves travel faster then light and i was told no. someone informed that the force of gravity itselft does travel faster then light though. soooo back to the origonal point of my question... if the force of gravity travels faster then light and we have a way of detecting small changes in gravitation pull, couldnt we take a given object in space, move it back and forth rapidly and have an observer detect intantanious changes in gravity, and communicate faster then light that way? i spend a lot of time thinking about these sorts of things.