The following page has a graph of Full Load Amps vs voltage (see Figure 2 about 1/2 way down the page):
The graph shows that F.L.A. is usually at a minimum at (or slightly above) nominal voltage.
As a first order model, the speed is approximately constant with voltage and load (since the speed changes only at most a few percent over normal ranges). If the speed doesn't change, then the torque cannot change, because the motor must match the load's torque-speed curve for the constant speed. Since power is the product of torque and speed, power is approximately constant. Neglecting power factor variations, the product of curent and voltage will also be constant so current must increase as voltage decreases below nominal.
But why does current also increase when voltage increases significantly above nominal? Because the assumption of constant power factor doesn't hold as we increase voltage significantly above nominal.... the core will go into saturation and draw extra inductive current (which doesn't contribute to real power).
Another refinement to the model... if you watch closely as you decrease voltage you would actually see speed decreasing slightly. The reason is that the torque speed curve in the operating range shifts down as voltage decreases. Accordingly, the running torque will likely be slightly less also.
All above discussion pertained to running behavior. For starting, starting current is proportional to voltage, and starting torque is proportional to voltage squared.