Sorry for the delay....busy.
There is no way to keep the screen clean while washing. You have to rinse it frequently to keep the debris and wood fibers from bonding to it though.
Rinse at an angle though from both sides. Elongated wood fibers tend to lap over the individual wires. Rinsing from an angle helps to get them loose better.
Chemicals used on the trim covering screen sections will damage the screen and remove the coating turning them shiney or more like bare aluminum. If the screens are fiberglass, they can be damaged by the chemicals and during the wash actually tear.
We often recommend just replacing them instead and applying a coating that will last a long time inside the porch to minimize the need for future cleaning.
Rod