How do I exit the window manager on the current X display?

Let's say I have the window manager Blackbox running on two different X displays, e.g. :0.0 and :1.0 launched from tty1 and tty2 respectively. From a terminal (emulator), how do I exit (kill) the Blackbox process that uses the current X display? The command killall blackbox does not fit since it terminates both sessions.