It depends on the digital mixer. Many digital mixers run on firmware rather than software and such firmware is much simpler than a modern operating system, so sudden loss of power is not a concern due to there not being any critical state data that could get lost. The board will start up clean from a fixed state every boot regardless of what you do. There might be issues if you pulled power in the middle of saving a preset or something, but otherwise it would be fine.
Other boards actually are just USB control surfaces to an actual computer, some even running versions of Windows. These consoles are a bit more likely to have problems as a result of sudden power loss since Windows might do something under the hood that you don't realize it is doing. The best practice will be to follow the documentation for your board on these. I don't expect a major problem is likely from sudden power loss as not a lot is happening in terms of persistent state changes that might be in-process (and even desktops are a lot more resilient to that kind of thing now), but if the manual suggests a particular power down procedure, it should be followed.
Keep in mind that both of these still should follow the rest of proper system shutdown, with shutting down from output backwards (speakers to board) to avoid spikes and pops through the system as components may still surge slightly on power down.