we have had dollies like these, but here's the thing. The places we had to move from and to had to involve, at some point, a passenger elevator. That is an elevator designed to take people and not freight. One of the main differences as that passenger elevators have narrow doors. They will not accommodate a map case sitting flat. They are even too narrow for drawers sitting flat on a flat cart.
We had to take down the cases, remove the drawers, empty them, ideally. Sometimes we had to put the drawers on their sides with the maps in them. In those situations, the drawers should be numbered so they go back in order!
The biggest thing I recommend is to respect the crew who do the heavy lifting. Getting acquainted with them is step one. Step 2 is to lay out what you want to do, and how you want it done. Step 3 is to stay with them throughout the moving process.
This paid off for me many times over. I have had to move map cases several times. Several of the same people have been involved, and they seemed to enjoy working with me as much as I did with them. They took very good care of my maps and equipment during each move I have done. To you those maps are your babies. If you work closely with the crew, they'll make them their babies, too.
Ken Grabach
Maps Librarian
Miami University Libraries
Oxford, OH 45056 USA