Here’s a tip from Chris Peeples at Vixen Hill for cleaning Porch or Gazebo screens:
I’ve always used one of those soft bristle brushes on an extend-o-pole. They’re generally sold for washing windows. Dip the brush into a hot bucket of water spiced with a little dish detergent and lightly brush both sides. Then, hose out screens from the inside and mop up any excess over-spray.