I just replaced the doggy door on our screened-in back porch. When we had the porch re-screened a couple of years ago, I asked the workmen to add a horizontal crossbar between two of the vertical 2x4s, and just screen that section upwards from the crossbar. Then I made a flat panel of vinyl screening (the new screen is metal, but we had some vinyl left over from when the house was built) to fit the opening, with double side and lower hems, drapery weights in the lower corners, and wide velcro sewn across the top, to hook onto velcro tacked to the crossbar. Vinyl screen is easy to sew on, I discovered.
The dogs love having their own entrance, and it sure saves us a lot of aggravation. After two years of heavy traffic, though, the panel was getting frazzled and it was time to make a new one. I didn't want to buy new velcro but dreaded having to pick out the stitching that held it to the screening; however, once started, it ripped out easily.
Sorry the inside picture is so washed out...I had to lighten it considerably to show any detail at all.