The type of glue really depends on the material being used. When you say "fabric" mesh, you may be referring to natural materials (cotton, for instance), a synthetic (nylon), or perhaps material which is like the fine mesh materials Tamiya puts into some of its armor kits for screening (polystyrene).
With polystyrene, use liquid cement like Tamiya extra thin. This is so thin that you'll have to go out of your way to apply dozens of layers to begin filling in the squares of the mesh.
With natural materials, craft glues like Aileen's Tacky or even standard white PVA glue like Elmer's work fine to bind the fabric to the plastic, provided that you are not relying on the glue for strength or structural integrity. This will plug up the holes if you get sloppy, but work slowly and carefully to get the best results. In the case of some synthetic nylons, the same type of glue can be used or epoxy can be substituted for nylons that refuse to bind or where you do need extra strength.
Airbrushing over mesh is easy enough. The only time you'll run into issues clogging up the mesh is if you're hand-brushing and applying too much thick paint. Thin the paint and use very light coats.
Matt issues are easily resolved with the appropriate clear sealer, such as Dullcote.