Drain current of NMOS circuit is constantly changing on DMM
FETS have temperature coefficients for Vgs versus Ids.
Thus heating changes Vgs and causes Ids to also change.

The usual method of measuring IV characteristics uses a voltage supply for  Vgs, and a 2nd voltage supply+current meter for Ids.

To  measure at high currents (or high power), use a "pulse" test setup. Tektronix 576 curvetracers has popular.