need advice for glue

If your doing frame and panel you don't glue the panel in place,because of wood movement.

Jim these are the inserts behind the sides. Hide the kreg screws and fill in the spaces for a 3D look. should be no movement. usually the tray shelf hold it all tightly and will on top this time but the bottom flexes. ??

Solid wood always moves by taking on moisture or drying out . If it's ply that's another matter.
I would guess the bottoms flexing because of wood movement, did it bow when you first put it in? How would you feel about brass screws with elongated screw holes.

Jim, I went with the glue on the ply. Answered my own question, does NOT stick well on a polyurethane surface. But did stick long enough to get my screws in holding all in place flat and smooth now. I had no problems with the bottoms just a piece or two of the birch 1/8 in plywood. I drilled the shelf rests and counter sunk the holes to hide the screws and this worked great. Thanks for taking the time to assist me in this, I appreciate it. You should have heard those glued panels popping as the screws drew the birch ply down flat.