Do other 3421A users see this behavior? (Or users of other high impedance voltmeters?)
If the unit is set to DC volts, autorange with the input open circuit, and the controller asks for readings over and over, the unit returns readings that slowly increase from 0 to .3V, then it switches to the 3V range where the readings continue to increase until the unit switches to the 30V range. At this point, because of the lower input resistance (10 Mohm), the voltage drops to near zero and so the unit switches back to the .3V range and starts over. It does this even with no mux installed and the internal harness unplugged. When I connect a 100Mohm resistor (the largest I have - I don't even know how close it is because I can't measure it!) to the front panel jacks, the unit reads zero all the time. BTW, if the unit is set to .3V or 3V (not autorange), the reading rises to overrange and stays there.
My theory is that this is caused by leakage from the coils of the relays that route the signals on the main board. The coils are all connected to +5V and the other ends are switched to ground to actuate them (they are the same latching relays that used are on the mux board). The specified input resistance of the voltmeter is 10 Gohm (10^10 ohms!) on the two low ranges and I imagine the input capacitance is being charged from +5V through the leakage resistance of the relays. I think this is the kind of leakage that is supposed to be avoided by the "guarding" techniques recommended in app notes for high input impedance amplifier ICs. The relays probably don't have any provision for guarding (would require an extra pin to be connected to the guarding potential).
If I ever get very brave I might try to isolate the relays from +5V to test my theory (since they are latching relays they don't need power to stay in their current state).