There are highly-insured companies who specialize in moving art and installing it.

I know this is the company that many area museums use: http://www.artechseattle.com/

In the event of moving very valuable art (such as the Picasso show), usually a curator or two will accompany the art on its journey, and then stay in the city where the exhibition is up for the duration of the exhibition. I know that when SAM had the Roman statuary show from the Louvre a few years back, that two or three curators accompanied the show AND the contract with SAM required that there be a guard with the pieces 24 hours a day. I would not be surprised if this were the case with the Picasso show, as well.

As for moving the objects themselves, basically it's the same as packing and moving anything: you wrap it really well with protective material, put it in a large box or crate (sometime custom-built), and then you move it very carefully with a very well-trained staff of movers. I remember several years ago I poked my head into a few galleries that were being renovated at the Tate Modern; there were priceless (and very famous) paintings wrapped in bubble wrap, just leaning against the walls. In particular, I remember seeing the famous John Williams Waterhouse painting of the Lady of Shallot wrapped in plastic. It was a little surreal, like seeing someone famous in their pajamas.