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.