For Constant voltage power supply, the current will change accordingly when the load is changed.

For Constant current power supply, the voltage will change accordingly when the load is changed.

LED is a kind of diode, when there is a little change in voltage, the current go through LED will have a comparatively bigger change, which will change the wattage accordingly. The i-u curve of LED will change when the temperature change. When the temperature arise, it will decrease the forward voltage of LED, or we can say, the voltage on the power supply will be bigger, which will result in rising of current which finally make the wattage increase.