Transistor Switch to Pull to Ground (Broadcast GPIO)
Since the Sonifex can be driven by 5 volt logic, you should be able to connect your logic output directly to the input lines, without the transistor.  You can add the transistor if you need a logic inversion, and/or if it makes you "feel safer".

In any case, your logic ground must be connected to the Sonifex ground.

If there is any problem connecting the grounds, you should use a reed relay or opto-isolater to provide galvanic isolation.