Kleinstein, AVGresponding:
What I expect to be inside the multimeters is a voltage limited, step controlled current source. The current is controlled (influence measurement) and different on many ranges, while the voltage is just limited (by giving a limited voltage input to the current source) and has not any requirement for accuracy or stability (in the 34401A the voltage is derived from +18V power with a simple 5.1V zener, I don't think in the U1401B took care of deriving it from a reference).
Unless there is a reason (like not to push semiconductors in conduction on some ranges, or the opposite) I expect it to be the same for all ranges even if might seem to have changed, in case multimeter insert a parallel resistor in highest range.
About falling time, if total capacitance is 10nF and total "leakage" resistance is 10Gohm, we would have a tau of 100s. Long, in effect, but I suspect both values being lower, in particular capacitance, so you should see voltage to settle in a reasonable time, if it is really stabilized.
It would be interesting to test again the U1401B with different condition batteries, to see if and how much the testing voltage change, as well as in the last but one range.