Use the internal FileManager of uTorrent 2.x or 3.x (not 1.x).
Then uTorrent is able to keep track on where the stuff stays and keeps working.
When you move things to another PC, then you should make uTorrent 'portable' first.
This requires 30 seconds downtime (when you know what to do).
Be carefull with this, otherwise you have to reload (and recheck) every torrent, loosing uTorrent' statistics (not BakaBT' statistics) in the process.
So yes, you can move things while uTorrent is running, when you use the internal uTorrent file manager.