I'd try a brush adapter for a drill, soft bristles and run it on slow speed to loosen up the stuff in the cloth, then use a vacuum to suck out the loose stuff. They also make the dust wands in long lengths that should be enough to get into the case.
https://www.homedepot.com/p/Ettore-Poly-Fiber-Professional-Cobweb-Duster-31028/204267870
https://www.homedepot.com/p/Swiffer-Dusters-Extender-Starter-Kit-003700086554/202828154
https://www.homedepot.com/p/Ettore-Poly-Fiber-Professional-Cobweb-Duster-31028/204267870
https://www.homedepot.com/p/Swiffer-Dusters-Extender-Starter-Kit-003700086554/202828154