As for usage, it is very easy:

Suspend the respective process with Ctrl-Z
Send the job to background using bg
Take away the ownership from the shell using disown
Start or enter your tmux/screen session
Run reptyr PID to attach the process to the current shell
It also has some additional useful features, such as TTY-stealing, which is documented in the man page.

Before compiling reptyr, make sure to check whether it is in you distributions repository. At the time of writing, this was at least the case for Fedora and Debian.

choli setup

Via blue iirc :
You can press ctrl-z to interrupt the process and then run bg to make it run in the background.

You can show a numbered list all processes backgrounded in this manner with jobs.

Then you can run disown %1 (replace 1 with the process number output by jobs) to detach the process from the terminal.

In spite of the name, the process will still be owned by you after running disown, it will just be detached from the terminal you started it in.