( How we do it, full version probably requires special permission )
Rtorrent by uses a session folder, to find yours, from the command line:
grep session ~/.rtorrent.rc
It is often ‘~/.session’
Inside the session folder you will find files with three extents, .torrent, .torrent.rtorrent, and .torrent.libtorrent_resume
The filename’s themselves match the hash of torrent of each of your loaded torrents. These files point to the location on disk or the downloaded payload, trackers, ratio data, etc.
Now the simplest thing to do, to move your torrents, is move the downloads to you new server, LFTP is ideal for this. Then Incrementally load the .torrent files from your old session folder into your new watch folder, say 10 every three minutes or so.
But if you want to retain your stats and avoid hashing, this is possible, move all the files in your session directory to your new session directory on your new server. Then create a facade using mkdir & symbolic linking, you need to have your downloads appear to be in the same directory as your old server.
Old Server Download Path: /home/u3/foobar/Downloads
New Server Download Path: /home/FooBar/Files
Now using mkdir (probably twice), create on your new server the empty path /home/u3/foobar (this may require superuser)
Then symbolically link new location to old location, that is Files to Downloads:
ln -s /home/FooBar/Files /home/u3/foobar/Downloads
You can now restart rtorrent, and all of your torrents with stats will be right there.