If you'd like to move a currently-running download, here's a way to do it on OS X (sorry if this is repeated elsewhere).
1) Quit transmission
2) Copy the download files and folders to the new location (e.g. to an external drive)
3) Edit '/Users/yourname/library/Application Support/Transmission/Transfers.plist' by double-clicking it. It will summon the 'Property List Editor'
4) Click the arrow by 'Root'
5) Click each of the arrows that show up and edit the value of 'DownloadFolder' so that it is the path to the folder where you have copied the existing downloads -- the folder that contains your downloads, not the download itself
6) When you are done editing all the 'DownloadFolder' attributes, then save from the File Menu and quit.
7) Start up Transmission again. It should pick up right where it left off.
