No, it is not constant current - both current and voltage vary with rotor speed. The rating - in either watts or amps - is usually a peak, and typically occurs at a high rotor speed, say 5000-7000 RPM.
The power that the alternator generates has to go some where, unless the circuit is opened (series style regulator).
There was a graph earlier on in this thread that showed how current generated varied with rotor speed.
In the case of turning your headlight on and off, lets say you have a 60 watt halogen bulb, draws 5 A at 12 VDC. At high revs, you have lots of excess power to run this bulb and say the other 2-3 amps of power consumers that you are running, say an excess of 2 amps or 24 watts.
Shutting the head lamp off under the same conditions, now you have 7 amps or 84 watts of excess generated power that needs to be dealt with. For the sake of simplicity, let's say that power and rotor temperature vary in a linear fashion - you now have 3.5 times the heat rise you would have with the headlight switched on. If the delta T were 25C per amp (plus the ambient inside the primary, lets say 80C from engine heat and a hot day at 30C), your stator temp at with a 7 amp consumed load would be about 130C, where as head lamp off with only a 2 amp consumed load the stator temp would be 255C. That is enough to char the potting on the stator.
Note that the delta T in this case is a contrived number to demonstrate a point. Please don't use this as an actual. But is is likely not too far off....
And this is only with a 10A (120 watt) alternator. Start adding more watts, and the issue gets a lot worse.