When I was working for the Osher Map Library we undertook a transition move of the collection during the expansion of the map libraries space 2006/8. The collection was moved back into the newly renovated and expanded space several years later. As the move was in house and not to a new building or off site we did not use an outside moving firm but moved the collection ourselves over several weeks. First we undertook a complete inventory and then moved the bound items as the interim space had shelving in place. Then we moved the flat files by removing the drawers with the maps in them, and then moved each case up to the new space by stack, and then the drawers so that nothing got out of order. We also did this with the oversize drawers, but this was however only possible as we had a large freight elevator in the building.

We did the move ourselves as we were happier moving the collection as control of thousands of valuable items stayed in our hands.

You would need to check with Ian Fowler the director of OML, on how they undertook the move back into the new space, but I think that they also did this without help. 

I hope that my thoughts are of some help.

George