Don't forget guys that they do have portable antigrav units and other exotic devices. Those cabs could have antigrav units to make them easy to move around and their own power and motive system (did they stop working when the ship was deprived of main power in "The Cage" or similar episodes?).
Good call! I dunno why that didn't occur to me. If these people have had access to reliable anti-gravity cars for the last 200 years (since c. 2036), why shouldn't a turbolift cab be essentially that? I like that idea a lot. It probably wouldn't have hardly any actual contact with the walls or floor of the actual tube, except possibly to "land" at a stop door or else have some sort of brake pads or clamps to hold it in place if it has to stop in the middle of a vertical tube (especially in an emergency, like in the TNG episode "Disaster" when Picard and the kids have to climb the lift shaft).
What a wonderful idea!