Got that right - electrical circuits will try to maintain power [watts] at a constant level. If voltage drops, amperage increases to maintain the wattage level.
I used to work for NEC in the Fiber Optic Transmission division. The designer of the power module for one of the fiber optic multiplexer models forgot to include an under-voltage cut-off circuit in the power module.
Note: The under-voltage cut-off is a mandatory BellCore spec.
Anyway, New York Telephone [now Verizon] performed the BellCore under-voltage protection test when it installed 100 of these multiplexers in a new Central Office installation.
They dialed down the voltage and ALL 100 power modules FRIED !!!
That's so true, the problem. A month ago I was designing a switching (DC-DC) power supply, and the low end of the input voltage range is the most troublesome. Not all circuits "will draw down more" but some do, if they are too smart for their own good. The regulator that I picked does have an internal cutoff at something like +3.6V, but that is to be tested once the board is back from assembly. Then I will know if it works or not :-)