What you want to do is impossible. The file formats are not the same on Desktop and Firefox for Android. In addition access to the user profile is on protected storage on Android which is impossible to view unless your device is rooted.
Ok. So this makes sense. The password manager only works for the exact site that set the password. If the website redirects to m.example.com and you have a password set for example.com the password is not shared.
You could go to the mobile site m.example.com for example and login to that site. This might require you to change the 'User Agent' to a mobile one on desktop.