With the CAGED system for minor seventh chords you can play any minor seventh chord in five different positions on the guitar fretboard. If you haven't already, please have a look at the introduction of the CAGED system.
The CAGED system for minor seventh chords uses the open minor seventh chord shapes Cm7, Am7, Gm7, Em7 and Dm7, which haven't been introduced so far.
Minor seventh chords contain a minor chord plus the minor seventh. They have much less tension than the (dominant) seventh chords and sound similar to the major seventh chords, only a bit more melancholic.
Here are the five minor seventh chord shapes combined on the fretboard:
Below is a list of chord diagrams that result when applying the five minor seventh chord shapes to the root notes C, D, E, F, G, A and B. Here is help on how to read chord diagrams.
Keep in mind that it's often possible to change the chord shapes slightly, e.g. by omitting duplicate notes or by adding an open string to the chord shape (for example, when using the D minor seventh shape to play an E minor seventh chord it's possible to add the low E string - this is indicated by the grayed out letter at the bottom of the chord diagram).
Also not shown below is that some barre chords can be played with the thumb over the neck instead of barring the strings with the index finger.