Voltages decreasing while using the 6 and half multimeter
Note 9 on page 12 of the manual states:

Specifications are for the voltage measured at the input terminals. The 1 mA test current is typical. Variation in the current
source will create some variation in the voltage drop across a diode junction.


Figure 1. The meters in question. Image taken from the manual.
I think you're seeing two issues:

While the meter is 6.5 digit, the current source stability is not. It will probably drift a little due to thermal effects.
1 mA and a diode Vf of 0.7 V gives a power dissipation of 0.7 mW in the diode's junction. This will cause a little heating and this will cause the Vf to drop. At some point the diode will reach equilibrium when the heat lost to ambient equals the power input.

You could check which is the larger effect by allowing the temperature to stabilise for some time before connecting the diode.