A pool table is a frame, rails, three slabs of slate, and felt. I think the "right" way to move it is to take it apart. When reassembling, the whole thing gets leveled and the slat slabs get aligned, with stuff to seal / smooth the seams. Then the felt gets replaced or reattaced and the rails get reattached.
If this is indeed the "right" process, its advantages are that a) slabs of slate weigh so darn much that it's hard enough to safely move one, let alone all three and b) it lets you get everything nice and flat and smooth after installing it.
I highly recommend you ask a real billiards store what they recommend. They regularly have "move and install table" as a service they provide. A local shop was actually quite willing to give me some DIY refelting advice.