As the voltage reduces the motor current will increase (but the speed and torque will decrease). Exactly what happens depends on the magnitude and duration of the dip. Limited decrease in voltages should not be a problems as long as the motor current remains below the nameplate rating. For larger voltage dips the motor may stall (or go into a starting condition), which would increase the current above normal conditions and require the protective devices to operate.
If your 20V drop is related to 400 V (just guessing), that's 5% which should not really cause problems. From your description, it may be that the overload protection is set at less than the name plate full load current. In that case you could increase it to overcome the issue.
If this control voltage is also derived from the system voltage, a drop in this may also cause the contactor to drop out.
If you have measured the voltage drop, can you also measure the current then you will know what is