What bodies in the wider universe are moving?
It's ALL moving!
...because space itself is expanding.
In addition to that, all of the galaxies are in whirling motion, the stars within them are in motion relative to all the other stars.
There is no body that you could say is at rest, and furthermore, there is no frame of reference that you could say represents absolute motionlessness. ( "There is no preferred inertial frame of reference", which was Einstein's way of saying the same thing.) Every body in the universe is in motion relative to every other body.
ALL of them! There are no "static" bodies in the universe.
Everything from the sub-atomic particles making up matter to the largest galaxies are in motion. A key to Einstein's General Theory of Relativity is that YOU can pick ANY point as your "stationary" reference point. Everything in the universe is moving relative to something else.