First off, not all motors are the same; AC or DC makes a difference, then within AC, you have Induction, Slip Ring Induction, Synchronous, Universal Motors, Switched Reluctance, single phase, 3 phase (even 2 phase), etc etc. Reducing the voltage does NOT always make a motor run slower, it depends on which TYPE of motor is being discussed and what the load is at that moment.
But speaking of AC induction motors, the most common type, the motor produces torque that spins the rotor and thus the load. The torque is created by the motor drawing power from the line and the torque created by the motor has to equal the torque required by the load. The amount of torque the motor CAN produce is a function of the correct ratio of voltage and frequency in the design of the motor windings. Any change in that ratio results in a change in the torque it produces. Too low of a ratio and the torque drops off, too high of a ratio and the windings saturate, producing more heat instead of torque. That ratio must be +-10% of that the motor was designed for in order to operate within specification tolerances. But in addition, the amount of torque the motor ACTUALLY produces is a function of the LOAD on the motor.
If you lower the voltage alone and keep the same frequency, the amount of peak torque the motor can produce will drop at the SQUARE of the change in voltage, i.e. at 80% voltage, the motor can only product 64% of rated torque at its peak. IF the load requires more than 64% of peak rated torque of the motor to accelerate or re-accelerate after a step change in lood, then the motor slows down. By slowing down, the "slip" of the motor increases, which causes it to draw more current in an attempt to get back to normal speed. But in drawing more current, that causes an increase in the internal losses in the motor and an increase in the heat it has to get rid of. If it can't get rid of that heat effectively, the motor burns up.
BUT IF at the time the voltage was dropped to 80%, the load was only requiring 50% of the motor's peak torque capability, the 64% that the motor is delivering is still more than adequate, so in reality the motor will NOT be overloaded and heat up.