In an electric motor, almost all types, the current is proportional to the torque.
If the load increases, in other words, the torque increases and the current will increase.
The magnetic field in a motor are related to the voltages.
The rotor produces a field that turns at the same speed as the supply. When there is no phase difference between the fields, the current is almost zero.
The phase difference between the stator and the rotor is called the torque angle. This is the load on the motor.
Changes in the magnitudes of the magnetic fields, caused by changes in the voltage magnitudes, have little effect on the torque angle. These changes mostly alter the apparent power and are therefore related to the power factor.
(This is a simple explanation and I am making it a community answer so that terminology can be improved by those that understand more than I do. :-)