You guys are actually both correct, and without knowing how the circuit board is set up I can't say for sure which is applicable for this application.
For many loads they will draw their rated power and is true of all almost all induction and synchronous motors. As voltage drops, current goes up to compensate. For purely resistive loads, if voltage drops the current will also drop. With electronic circuits, it could go either way. Many electronics are designed to use an input of anywhere from 48VAC to 240VAC and use a regulator built into the device to maintain constant power. The current going into the electronics will change based on voltage, but the output will be constant voltage and current regardless of what the input is doing.
If this is how the Kessil is set up, then yes, using a power supply that doesn't offer enough voltage can cause damage to the circuit board as it tries to regulate the output.