Easy way to transfer a private key to another machine?
ssh-copy-id makes it easy to transfer a public key to a remote machine.
Is there any equivalent tool for moving a private key?
update
Don't do this! (see answer)
Good practice says that you shouldn't copy private keys from one machine to another. When you need access from a new client machine, create a new key on that machine, and set up the corresponding public key in your ~/.ssh/authorized_keys file on the server.
Doing this allows you to disable access from one client machine at a time, if you need to (suppose a laptop was stolen, for example).