The 2000-SCAN card used in the 2000, 2001, 2002 meters is rated for < 1 uV contact potential per relay contact pair. That's not bad but not great.
I have a ... (ahem) few Keithley 7001 scanners that have model 7011-S scanner cards in them. They are rated for < 500 nV contact potential per relay contact pair, quite good performance for relays. Each card is actually four 1x10 scanners that can be used independently or combined (into 1x40, two 2x20, a 20x20 matrix, etc). If you are looking at switching a 10 V reference voltage to multiple meters, or multiple 10 V references into one meter, then you are only disturbing the voltage by less than 0.05 ppm.
Those relay based cards use latching relays to help avoid effects caused by coil heating.
Keithley also makes a nanovolt scanner card for the 7001/7002 systems that uses FETs instead of relays. It is an order of magnitude better but is very limited in voltage and current switching (10 V, 50 mA - and those are peak ratings, not average). That card will set you back about $3600.
(p.s. if you are interested in acquiring a 7001 scanner as described above let me know by PM)