First of all, load management systems for consumers are almost always voluntary and opt-in.
Second, I've never heard of a consumer load management system that dealt with dishwashers and washing machines - first, because these typically are minor power consumers compared to heating and air conditioning, and second, because yes, you shouldn't interrupt them mid-cycle.
It's not like demand management shuts down your entire house. That's more like a rolling brownout, which is precisely what demand management tries to avoid.
Maybe you're thinking of variable rates, which encourage people to use energy intensive devices when the rates are lower - like starting your electric dryer, which is a large consumer, when you go to bed at night, rather than at 3 PM in the middle of July.
It's amazing to me how people who have absolutely no idea how electricity works or how it is generated and delivered suddenly become experts on everything from RF radiation, privacy and security of IT systems, energy efficiency, and utility infrastructure.