I think the lighting guys regularly plug incandescent bulbs (floor lamps, etc) into dimmer channels, but it's these non-trivial electronic devices that seem to have the problems.
It's not that these devices draw much power that's the issue. A simple light bulb copes presents a reasonably linear load. These other things present inductive and/ or capacitive loads. These DO NOT cope with the chopped power waveform that dimmers present.
Get some long power cables and run your powered speakers from the same power points as the mixer, I'm willing to bet that will remove the noise, if it does, try lifting the signal earth at the input to the speaker or feed it through a transformer.You will get earth loop hums on the same circuits, as the earth wire
loops around the building, by plugging in at the sound desk you are star wiring the supply and can't get an earth loop.
Having a thought here. I'm hoping that the speakers are fed with a balanced line, is this the case? And yeah, at least try plugging the speakers to the same power supply as the mixer. If it fixes it, you now know what you need to resolve more permanently...