You can also do it using reredirect (https://github.com/jerome-pouiller/reredirect/).
The command bellow redirects the outputs (standard and error) of the process PID to FILE:
reredirect -m FILE PID
The README of reredirect also explains other interesting features: how to restore the original state of the process, how to redirect to another command or to redirect only stdout or stderr.
The tool also provides relink, a script allowing to redirect the outputs to the current terminal:
relink PID
relink PID | grep usefull_content
(reredirect seems to have same features than Dupx described in another answer but, it does not depend on Gdb).
