Does voltage fluctuation damage AV equipment slowly, and does warranty cover it?

Yes to the first, and mostly no to the second, which is why prevention is the whole conversation.

Damage first. Sustained over-voltage stresses the power supply cumulatively: capacitors and regulators run hotter than designed and age faster. Nothing dramatic happens on any given evening. Then eighteen months later the supply fails and everybody calls it unexplained. Surges are the fast version, and lightning does not do slow. Under-voltage is less destructive for modern switch-mode supplies, which tolerate a wide input range, but it causes shutdowns and reboots, and it strains anything with a lamp or a motor, so projector lamps and older gear feel it first.

Can you hear it happening? No. Can a service centre see it afterwards? Usually yes, which brings us to warranty.

Manufacturer warranties cover manufacturing defects, as per brand/distributor policy. Damage from surge, lightning or improper supply is commonly excluded, and it is not a grey area for the technician: burnt supply components and scorched tracks look nothing like a factory fault. So the practical answer is that the slow damage lands on you, and it lands at the worst time, when the equipment is old enough to be out of warranty and new enough that you are not ready to replace it.

Prevention costs a fraction of the repair. A real surge protector (a proper unit with a stated joule rating and a protection indicator, not a six-way extension board with a red light on it) for every rack. A stabilizer where your supply warrants it: if lights visibly dim and brighten through the evening, or you share a line with heavy motors, that is your answer. A UPS for a projector so the lamp gets its cooling cycle when the power cuts.

Rough supply where you live? Tell us when you buy. Protection is designed into the system at the start, in the rack, on the right circuits, rather than bolted on after the first casualty. The repair bill teaches the same lesson, only slower and more expensively.