The question not of a moving universe, but of a rotating universe (this is one we can test for) was adressed both in 1937 and 1949.
W.J. Van Stokum noticed a peculiar consequence of relativity. If there was a very dence infinitely long cylinder that was rotating and you traveled around the cylinder counter to it's rotation the whirlpool like warping of space, theorized by Einstien, would allow you to arrive back at your original starting point BEFORE YOU LEFT.
"That's great," you might say, "It's a realy handy use for all these infinitely long hyperdence rotating cylinders I have that are just lying around." It wasn't until many years later that Pysiisists discovered that you didn't nessesaraly need the cylindinder, if SPACE were rotating then the universe could act as one giant time machene. An object that leaves in a straight line would actualy spiral around and could return to any point in time at any point in space as long as it travels for long enough.
Unfortunately (or maybe fortunately) there is no evedence to suggest that the universe is rotating.
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