Lots of possibilities really. I’ve played around with a few.
There’s the ‘no-input’ approach which is maybe what sketchdashaman is referring to. It’s essentially just routing an output (usually an aux or secondary) of the mixer into an input (usually not xlr but 1/4). Analog is necessary and powered is best. This produces a feedback loop of the (usually) inaudible inherent electrical signal of the powered mixer. Twiddling the frequency knobs will begin to alter the sound and hitting mute buttons or volume cuts can produce rhythms (though you gotta do it manually). It is highly unpredictable and generally extremely atonal. Each mixer, even between the same brand and model, will have its own character and sound as well. It can produce damaging frequencies to speakers and ears, so using a limiter or compressor is recommended and headphones are NEVER to be used. Prolonged usage can also mess with the mixer, so its not really advisable to use a pricey one or something you plan on using for multiple uses. I have a Behringer that I’ve been using for years that can’t really function for anything else but this type of play at this point. You can run several mixers into each other and get some interesting internal modulations of the signals as well. But again, this is all highly unpredictable. You turn a mid knob, a high pitched squeal comes out. You turn the treble, you’re in throbbing deep bass or the sound just cuts out entirely.
Most folks who do this for a while start using effects to control and modulate the signals for a more ‘compositional’ approach. RIng modulators, reverb (particularly spring), delays, loopers, and filters are great for this.
Some folks use the mixer of old cassette recorders, both playing a cassette as a sort of sampler/instrument as well as feeding external signals through the cassette mixer. Some older machines because of age and neglect or maybe were always crap can alter and carve up sounds in peculiar ways. At some extreme levels and gain it can also sort of self resonate or sound like a heavily drenched kinda drone effect. A lot depends on the machine and source signal of course.
And yes, there’s just using mixers as they were intended but because of the machine itself (old, cheap, built in effects, unique circuitry or routing options, whatever) the sound produced is something nice, unique, odd, etc.
Digital mixers and automation is also something to consider. You can set up random parameter shifts, quick shifts in frequencies, frequency drop outs, gain staging, etc. Like sketchdashaman said, these are not particularly good at feedback though. You can get clipping, which on a drum track could be cool or maybe a kinda choppy effect on a lead but it could be tricky.