When you click to change to a new site the first thing bounSky does is pop the Lync / SfB client to the foreground and then send it a “sign out” command.
The client then shows on-screen with the SIP address and account details of the site you’ve just signed out of, not the one you have chosen to sign in as.
In the background bounSky pushes a sign-in request against the new site, but this isn’t visible to you on-screen.
I’ve tried but have not been able to push the new site’s details so they’ll show to you when this process kicks off. I appreciate this is often confusing for new users as it appears bounSky isn’t actually doing anything. It is – you just need to wait for the sign-in process to complete.