Folks,
The short answer is: Not well.
Dry paper and similar dry filter media stop large rocks and small babies, (figuratively speaking), the problem is they they have to physically block the particles, and the more efficient they are, the more the airflow is also blocked.
Some of you may know the "oiled paper/foam" type filters, ( K&N or similar) where the "sticky" oil holds a greater %, and smaller size of the grit in a (generally) synthetic paper/foam rubber primary filter. I have never seen this kind of filter with aviation approval.
Whether it is pumice, basaltic or other volcanic ejecta, they all make great grinding paste. I know, from fitting hot air filters, and oversized cold air filters to Ag. engines, what a difference it makes to engine life --- the barrels, pistons and rings --- and that was good old common or garden variety dust, benign compared to volcanic dust.
Few engines have hot air filters, so if carb anti-ice is required, you will probably be breathing unfiltered air.
Tootle pip!!