A frequent lack of understanding is 'how a motor tolerates being supplied with 208V service'.
The answer.... Is a little confusing. I'll try to make it clear.
First a definition. In this FAQ I refer to "dual-plated" motors. I'm not referring to 230V/460V by this, I'm referring to 208V/230V the motor may actually be a 208V/230V/460V motor. The 460V has no bearing on this discussion.
Historically motors in the USA were designed to run at 230V. The standards allow them to be fed plus or minus 10% of this 230V or something between 207V and 253V. However along came a new supply voltage 208V...
A lot of manufacturers of product's incorporating motors started getting calls about "but we only have 208V", "do we need a different motor?" There really weren't 208V motors being made so product makers asked motor makers this same question. The answer they cobbled-up was, "since 208V is more than the -10% limit of 207V, yes, you can run the motor or product on 208V.
However convenient this was it came with a problem. Remember motors were designed at 230V even though they were generally applied in 240V systems. This was a nod to the reality that the power sources often delivered less than 240 to motors. Voltage loss comes from lots of places like upstream transformers, and facility wiring runs. Hence the 230V "application" voltage and the generous -10% still being applied to that.
When a maker "re-plates' or 're-tags' a motor as "dual voltage" 208/240 they are completely trashing the aforementioned conservative approach. What this means to a user is if they are saddled with one of these dual voltage motors they have to realize that they MUST supply the motor with 207V and not a volt less. The +/-10% leniency is INVALID in this case. Their perfectly happy at 230V motor will be 'on the edge' of its voltage requirement. In 208V service the motor is really a -0% +20% motor.
Now when the facility supply has some voltage drops the motor will immediately find itself supplied with less than 207V. It is not uncommon to see 208V facilities delivering 200V or even 198V to their various motors. In my experience 208V facilities far more commonly deliver low voltage than the older 240V facilities. I'm not sure why but my guess would be because everything needs more current at the lower voltage to deliver the same power to all the various devices.
Since nothing is easy there's more to this. What a motor is driving is important to how well the motor will tolerate 208V service if it's a re-plated 230V motor. Cyclical loads like drill presses, punches, lathes, and any other load that varies frequently or continually usually fair well enough. This is because the motor doesn't actually have to deliver its full rated horsepower continuously. Your potentially undervolted motor will have frequent respite from the extra heating it will be subjected to due to low voltage.
Reminder: A motor is a dumb creature. It is designed to run a specific speed using the supplied voltage and demanded current. If the voltage drops more current is demanded to supply the same power.
Power = Volts x Amps --- Drop the voltage and the amps increases.
Back to the driven loads. If, instead of cyclical loads as mentioned above, the loads are continuous and at rated horsepower things will be very different. Loads of this type would be for example: air compressors that have to supply continuous air or pumps that have continuous flows. One of the worst cases are refrigeration compressors. A refrigeration system has the worst of it because they can be designed to run right at the motor's maximum ability and on hot days they can run at that loading for many hours straight. Even worse is on the hottest days they have to run in the hottest ambients and when the power company is supplying the very lowest voltage because their network is fully loaded too.
In these continuous loading applications you must be sure your dual voltage motor is absolutely positively getting 207V or more or your motors WILL FAIL. If you have a dual-voltage motor you would do well to include boost transformers to get the voltage way higher than 208V, so in the event of a sagging supply your hard working motors will never dip below 208V. Your motors will generally last longer and run cooler if you provide them with higher voltages than the minimums.
One more complexity. Fairly recently, as in the past couple of years, product makers and motor suppliers have started to standardize motors to cover more of the world with less motors. To do this they've created a class of motor that's actually designed as a 200V motor not a 230V motor. These motors will be very happy at 208V or 207 or a sagging 198V as the +/- 10% associated with them easily embraces the 208V. These are not dual-plated motors.
With a dual-voltage motor keep the voltage at the motor at 208V or more if the motor could potentially need to supply it's rated power. If you can't boost the voltage then you need to help the motor in other ways. Reduce the load. Do this by reducing the flow, reducing head pressure, using a smaller pulley on the motor, or restrict loaded periods. Also consider adding cooling air flow or reducing the ambient somehow.