Moonlight only supports Xbox controllers or making a different controller (like a PS5 controller) appear to Windows to be an Xbox controller. You can't use a PS5 controller natively, instead windows sees the ps5 controller as an Xbox 360 controller.

It's fine for most cases but not for motion controls.

Not GP, but I use virtualhere to forward my steam controller over LAN to my desktop. Moonlight handles controllers fine, but it's forwarding button presses, not forwarding the USB device. This rarely makes a difference unless you're using peripherals that need software.