Not an expert nor an EE, but here my take.
Constant power (CP) load usually more complicated than constant current dummy load, and also "again" usually came with certain protection mechanisms like cut off whenever reaching certain voltage or current, this is to protect the power source from over drained or over current/temperature and also a timer. Ideally its also equipped with data logging feature to measure the watt/hour.
Remember, constant power means whenever the voltage is lower, it will drain higher current in order to get the constant power and vice versa. (Watt = Volt x Amp)
Examples, using it on Li-Ion battery, the purpose is to see whether the battery capable of sourcing "constant wattage" for certain load level, say like a cellular phone which has a switching boost circuit in it to supply a constant voltage to the circuit. So the CP device will start to drain the battery to simulate the cell phone, and as we know the battery voltage will drop when draining started while the power switching circuit will eventually starting to draw higher and higher current as the battery voltage is getting lower to compensate the required constant voltage for the cell phone's circuit.
The extra protection mechanism will kick in when the battery voltage reached the minimum safe limit before damaging the battery or certain maximum current, and sometimes certain max temperature as well. Once done, the results from logged data + the timing will show the battery capability to give the required "constant" power for certain period. Thats how they came out the popular term called "talk time" for cell phone.

Hope this helps.