Well, you won't burn out your alt, and you won't damage your equipment, you will just get less power.
If anything, I would THINK that with a serious voltage drop the biggest concern is going to be problems to the vehicle itself. I know dropping to 13v won't do much, but you may see your lights dim. You start dropping below that/12v, I could only guess you might cause a sensor fault from lack of voltage, might cause problems to your ignition including a misfire from coils not sending a full spark. You drop it low enough you could kill the battery if you pull too much voltage and are still trying to keep the stereo up. Amps have a working voltage, so if you drop below that, I would only guess it would go into protect.
I'm surprised nobody has a better answer though.