I am trying to move small collections into larger collection sets and it takes several minutes for each move to be completed. The files I am moving are generally pretty small so I don't quite understand why it would be taking so long. Is there any way to speed up the process?
This has nothing to do with file size.
It is all done in the catalog, the only thing I can recommend is File->Optimize Catalog.
DJ probably has you pointed in the right direction.
You are not moving files. You are just updating the catalog association between the files and the collections. It is all just database updates.
How big is your catalog (*.LrCat file)?
How many photos are "in" it?