How to (safely) produce flying dust on set?
I would avoid burning things to make ashes. Not very environment-friendly.
Instead of paying attention to the acting and to the light, and framing you will be worried about the effects. That is not a good idea.
Now days is pretty simple making particles in a 3D program. Try Blender, that can track the footage and have a very robust particle system and render options.
Here is an example of how a normal scene can be transformed into an apocalyptic scene, one typical example is Silent Hill.