Oscillating voltage in Wheatstone bridge
Typical DMMs use reasonably high input impedances between 1-10MOhm but source values approaching this would make for slower settling times.   

More likely the contact resistance is changing and swamps the surface resistance reading.

The use of 4 point probing may be indicated. https://en.wikipedia.org/wiki/Four-terminal_sensing

The Wheatstone bridge (invented for nulling measurements) is not often required in the modern day unless you are measuring small changes in bridge resistances like opposing changes in strain gauges and such.  The resistance bridge works best when balanced so not ideal for automated measurement of unknown resistance values.