Hi sydneykotic, when you say you 'move from place to place', does this mean that your computer will be moving with you?, if so, then just boot it up when you get to your new location - and hopefully, WCG will then be in a posiiton to receive your completed WU's.
If your move means that you'll be leaving your kit behind (say, you're moving from one employer to another), then I think (depending on when you move/when WCG start allowing WU's to send their data back), the data may be lost - as I don't think it's easily possible to transfer WU's from 1 machine to another. I think the software checks the compuer name/OIS etc. My suggestion (if it is this, latter scenario), woiuld be to try and leave it as long as you can - and hopefully, the completed work will get sent back before you have to shut down that computer.

You can simply take the whole BOINC data dir with you and use it on an other computer (if that computer isn't too different). At least on Windows it's pretty simple, simply install BOINC and point the installer to the copied dir as data dir. Not sure about Linux, but I guess it should work there somehow too, the way TonyEllis has described it.

Would guess even under linux BOINC stores client_state.xml under the "root" /boinc/ directory. Since you can't upload anything without client_state.xml, would expect you'll need to backup everything in /var/lib/boinc/ + sub-directories.

For Windows users it's normally c:\programdata\boinc\ + sub-directories that needs to be backed up.