It is the same for all clients as the all use TCP/IP connections.
So if you entersd a static IP address in your X-Box menu, you also entered a gateway and DNS server.
Depending on the usage of DNS relay, you can either enter the IP of the router (DNS relay on) or you have to enter an IP address of an external DNS server (you'll find 2 valid addresses in the WAN status section of the router's menu).
They all behave the same.