I use Apple Barrel and Folk Art paints, which aren't specifically designed for air brushes. From what I've learned here, that means there may be larger pigment particles, which clog the small passages in an air brush.
Because I'm lazy, and only paint for myself and friends, I thin those paints with Auto Aire 4011 reducer until they are like skim milk before I use them.
When I do get a clog, I usually just backflush the paint back into the cup a little to clear it. If that doesn't work, I pull the needle back a little to make more room for the paint to pass. It's not scientific, but it works most of the time.
If I were going to screen the paint, I would thin it first, because it comes out of the plastic bottle way too thick to use anyway.