I don't believe that it's going to kill anything, but I could be wrong. As far as I know, there is a minimum voltage that is needed, and if it's not there, the light won't come on, and no current will be drawn.
The CC supply shouldn't die running undervolted, simply because it is given less power, therefore there's less power to damage it. It will need a minimum voltage to work though, and if it doesn't see it, then it just won't do anything.
I would think that, at the very best, you might be able to give the CC a variable glow (as is now available on the market).
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