Ol' fashioned tube amps without any modern stuff inside 'em would probably have no problem other than diminished power & maybe sludgy tone during brownout periods. Anything that depends on a voltage regulator would be in deep doodoo. Most solid state preamps as well as built-in effects devices dependent on the usual plus & minus 15V supplies, would suffer having their power supply regulators quit during dips in power.
Also amps and other items using switching power supplies, can't count on those supplies operating on low line voltage. In many cases when the power line dips, the power supply goes on vacation.
Way before the digital age, I participated in a medium size outdoor fest where the whole PA system cut out on every kick drum hit. How embarrassing! They were using dbx160 compressors as limiters across the house mix. Later I found those classic compressors, while wonderful in every aspect, had muting circuits that clamped the audio output to zero if the AC line wasn't up to par. When the power amps drew a lot of current to amplify the kick drum, with each hit the line voltage dropped for a few milliseconds. Had the PA outfit known that, they could have snipped a couple of FET's out of each compressor's mute circuit and avoided the problem permanently. Live and learn!