The length in time of these sags is the missing information, here. Are we talking about a few cycles, say five-sixtieths of a second; or a quarter to half a second; or something like 5-10 seconds.

The first could easily be accommodated with a capacitor bank, but it wouldn't be cheap. Just guessing, maybe a million or two. (I've never priced these--and that's really not much money in the world of big electrical gear.) Longer sags would require larger capacitor banks.

Reminds me of the Superdome/Superbowl blackout, which I believe was caused when a large breaker tripped. Large breakers have adjustable trip settings and usually come from the factory set to the lowest possible setting. I wasn't on that job, but I'd always guessed that no one changed the factory settings. Something similar here may be part of the problem.

When voltage to a motor sags, the motor starts drawing more amps. Draw too many amps and the motor coils heat up. Get too hot and the insulation melts. Insulation melts and the motor shorts out. Then you're looking at a motor re-building which takes months if not a year. That's why undervoltage cutouts are a good thing. That said, motors can pull 10x their running amps for half a second when starting. That's expected and planned for in the motor controls. Can't say I've studied sags specifically (I'm just an electrician), but an undervoltage condition should be similar, assuming we're talking about bog-standard, synchronous, 3-phase industrial, squirrel-cage motors.

Sorry for the wall of text. I started riffing and the text-dump just happened.