Wondered this today whilst walking around the ancient Egypt exhibition at the British Museum. Do they move around the globe a lot? Some of the pieces such as pillars and statues are easily 20ft tall so I imagine it would be a big and delicate task.
It's accepted best practice to try and move most artifacts as little as possible. Generally it would just be between exhibit and storage. Museums do often loan objects to each other, which is a process involving significant amounts of paperwork, but it is an excellent use of collections.
Larger pieces move much more seldom. The National Museum of the American Indian has several 20-25ish foot tall totem poles in storage, that actually go up between floors. There was some special crane system they used to get them there, and in all likelihood, it would take a lot of motivation to get them moved again, even for an exhibit.
Smaller things like statues travel more often. There are companies of shippers that deal specifically in art or artifacts. Often registrars or collections managers also curious the artwork to its final destination, completing condition reports when it is packed and when it is unpacked.
The hope diamond was sent from New York to Washington D.C. via registered mail.