Does it kill an A/C to run it off a generator?

I forgot to add that when testing one of my outlets as per a DMM, I get 120 VAC when the A/C is on, and 122 VAC when the A/C is off. There is a brief transition, but I've seen the DMM hit ~105 or so for a fraction of a second, but not lower.

Subsequent compressor starts/stops don't bog the compressor nor drop the voltage under 115, until the next day.