Is spacetime moving in general relativity?
If not, how does spacetime retain its past, while moving toward future?
Spactime doesn't "move", the idea doesn't make sense. Spacetime is what things move in. The "moving towards future" is entirely psychological, it's all in your head. It is part of the map between the physics and experience, it is not part of the physics. It hasn't been part of physics since space-time became a concept.
The meaning of "it's all in your head" means that you need to carefully formulate what it means that "time moves forward". In terms of experience, it says that at a certain time, a person can write down their memories, but not memories of the future. This is included in the physics, since you can simulate a person, and see them write down things from their memories, and not things from the future (assuming entropy increases into the future). Any which way you give precise meaning to the statement "time keeps getting pushed forward", the meaning ends up in measurements on psychological states, and it has nothing to do with the structure of the laws of physics.
So it doesn't matter one bit whether you consider space-time to "exist all at once" or "push itself into existence bit by bit through time". These two positions are identical in the sense of positivism. This is another place where positivism is essential to physics.
When any superficially different answers to a given question cannot be distinguished by experiment or experience, the question is meaningless, it is a self-deception using language, like the question "Are S-shell electrons happy?" There is non-psychological test of whether time is pushing forward or not, so the question is really your brain fooling you.
I believe it is better to say that waves, curving, and other distortions in spacetime travel through spacetime because you really don't want to give the time axis preference in GR. GR was invented, partly, in order to explain such phenomenon- and therefore abolish gravitational action at a distance. If gravitational wave detectors were more powerful, scientists could observe objects oscillating in the lab. The oscillations probably shouldn't be described as "spacetime moving," but rather as, oscillations of spacetime because saying "spacetime is moving" implies that the entire framework itself is moving and then what is it moving through?
In frame dragging, spacetime moves.
Dr. Erricos Pavlis of the Joint Center for Earth System Technology says: "General relativity predicts that massive rotating objects should drag space-time around themselves as they rotate, ... Frame dragging is like what happens if a bowling ball spins in a thick fluid such as molasses. As the ball spins, it pulls the molasses around itself. Anything stuck in the molasses will also move around the ball." http://www.phy.duke.edu/~kolena/framedrag.html
The phenomenon is nicely summed up in this 86-second movie: http://einstein.stanford.edu/Media/FD_Measurement-flash.html
As the system evolves through time, its past is retained in the 4-momentum of its parts.