Isolation transformers would would be bulletproof, no?
Isolation transformers wouldn't help you here. If your problem was earth loops and hum, then yes, but as described here, no. Impedence effects on transformers get reflected input to output, and output to input.
A decent mic amp should have a reasonably consistent input impedance as the gain is changed. If it hasn't, then changing the gain on one desk could affect the level received by the other desks.
However, if you are saying that a third desk was introduced somewhere between your sound check, and the performance - that would certainly affect the level received by the existing desks. Eack new desk loads down the mic signal. It may not be a lot (depends upon the desk), but it likely to be noticeable. A mic is (say) a 200 ohm source, and you are adding three (say) 5k loads to it - these loads appear in parallel (a classic potential divider) - the more desks you add, the more it loads the mic. If any particular load (desk) is less than the others, it will have more effect (so the desk is a variable), if the mic is more than 200R, it will have more effect (so the mic will be a variable).
It might also change the sound of the mic (recall the current fashion for putting variable impedance impedence on up-market mic pres, for the 'subtle' effect it has on the sound?).
Q. Were all these desks supplying phantom? (Ok, don't answer that!)
Alan