Yes, unquestionably FedEx acquire and dispatch USPS box. FedEx is working independent of all the post offices and has initiated buying courier companies.
FedEx ships every box which is labelled right due to which it often accept other companies packages and ship them. So, yes FedEx has shipped USPS boxes and handle interstate deliveries of USPS network too.
All shipping of USPS is done by FedEx which crosses the state lines which make it more economical for USPS. It’s not at all same in case of USPS as it cannot ship anything labelled with FedEx.