Mesa 7i76 power

in the manual for the mesa 7i76 it says

CAUTION: VFIELD Must be connected directly to the DC power source with no
switches, circuit breakers or relay contacts in the circuit. A fuse is acceptable but
in no case should VFIELD be switched ON through a mechanical contact.

just out of curiosity, what's the reason for that?

If directly connected to a power supply, would it cause any problems as the voltage drops off when the power supply shuts down, or can the board handle that?

Electronics that drive loads, especially inductive loads, do not like to be switched at the DC side as that causes back EMF or induction spikes that can ruin the drives/electronics.
Mesa drives such loads like relays and solenoids and whatever might be connected to it's outputs....
In general it is never a good idea to switch the DC side of things, except when it is, and explaining all that takes some effort.

Its also that power electronics (and power supplies) have filter and bypass capacitors.
Connecting a charged capacitor (in the power supply) to a uncharged capacitor (in the
powered device) directly via a switch or relay results in a huge current surge (possible 100s
of Amps) for a few microseconds. This current surge can damage components directly
and indirectly by causing voltage surges due to stray wiring inductance.