Look at the pictures in the section labeled Drawer Slides in
www.plesums.com/wood/bedroom/drwhite.html
This chest did not have dust panels, but the construction would be the same, with a groove in the middle of the slides and the front and back, to capture a thin plywood or masonite or whatever dust panel.
I glue the front in place, made of premium wood, and matched to the "show" wood of the piece.
The sides, which also function as drawer slides, are 1/8 inch thicker than the front and back (1/16 above, 1/16 below) so the drawer doesn't ride on the front show piece and wear the finish, and so the drawer doesn't catch on the back.
The sides/slides are NOT glued in place, so they can slide forward and back in the dado or, in this case, the sliding dovetail. A dry mortise and tenon in the front and back add strength in addition to the dovetail or dado.
By leaving the sides completely floating, with a gap to allow the cabinet sides to expand and contract (with hardwood sides, in some seasons the back of the cabinet will be closer to the front of the cabinet that it is at other times of the year), I can then glue the back cross piece in place, so I have a fixed place to attach the back.
With fancy cabinets, I use shiplap boards, about 3/8 inch thick for the back, nailed in place (with an expansion gap between each board) with rectangular nails. Those nice rectangular steel nails come from my brad nail gun. With simpler cabinets, I just use 1/4 inch plywood back, but nail it to each cross board so it doesn't rattle.