It's quite possible and even easy
Screen command will do what you want. Run screen, start process, detach the screen with CTRL+A D. To re-attach run screen -x. You can also have several processes running under screen, create a new process with CTRL+A C and use CTRL+A with numbers to switch between running processes. Man screen. It doesn't really "move the process to other shell", but the effect is the same. The process will keep running (but out of sight) even if you log out.
I always have a screen running at one of the machines of my university. Keeps me connected to irc and present on the channels I want to be present at all times. When I re-attach the screen I'm also able to see what's been going on on those channels