How does dumping excess voltage works?
You are right to think that this type of protection depends on the clamping voltage supply to sink current.

Normally, this kind of clamp is used in conjunction with some sort of series impedance between the signal source and the node being protected, selected so that it has minimal effect on the signal under normal conditions, but limits the amount of current through the diode to an appropriate level under fault conditions.

Most power supply regulators cannot sink current at all, unless they're specifically designed for it. Instead, we rely on the other loads connected to the supply to sink the current, which means that any current flowing through the protection diode simply reduces the current being drawn from the regulator. If the diode current exceeds the load current, then the regulator current will drop to zero and you'll lose regulation.