Why does the value increase constantly and slowly in diode mode?
I wonder what they might have connected on the circuit in parallel with the diode


Capacitor or RC circuit is connected in parallel with the diode.


  Why do you think I see a constantly increasing voltage value when I change the probes?


Diode measurement operates using small current. It takes time to charge the capacitor, that is why you see such behavior.

By the way I tested with my Fluke, added 1000uF capacitor (first pick up) in parallel to diode and indeed it takes time until reverse-diode measurement shows open circuit.