Induction motor

Induction motor is constant power machine current increases as voltage drops, also the mechanical power developed (torque x speed) is also reduced, the remaining power is simply lost as heat. Increased current increases copper losses.
So voltage drop reduces speed of motor not only that it also reduces its efficiency.by developing lesser mechanical power.

If we consider other scenarios of load torques, for fan type load torque decreases with speed, which will make power output even lesser, and probably the input too reduces if the torque drops too sharply with speed. For constant power type motor load torque is actually more at lesser speed, and I believe it is only here you have some chance of getting same or even higher power output at lower speed (even though of course power input required here too increases dramatically)

"An AC induction motor is supposed to be a constant power motor"
I guess this is for frequency control and it means that the maximum output power (torque*speed) is constant regardless of the speed (for high enough speeds). This is just a kind of rule of thumb to know what the maximum torque vs speed characteristic of the motor looks like.

There will be little fall in the running torque supplied by the motor within the working voltage range of the motor and so the mechanical output and hence the electrical input remains practically same. The constant power concept is for practical purposes, as a thumb rule for frequency control like by variable speed drive. (In fact it is also a major assumption for load flow studies to take motor load as constant power.)