Speaking as an iOS developer, the issue that some of you are running into with browsers on mobile (both iOS and Android) is that the player that Broadcastify is using has to constantly request updates from Broadcastify's servers via JavaScript calls. When you background your browser, these calls are going to get paused because, let's just say it'd be bad if your browser could just randomly fire up and start making JS calls in the background. I mean Google's ad business would love it, but us consumers and govt. regulatory bodies, not so much.
There's no real work around for this (besides always keeping your phone unlocked and your browser in the foreground) as it's intended behavior.
That being said, if Lindsay wants to really bloat his AWS bill and start manually merging playlist calls into a live streaming playlists, that'd get handed off just like any other kind of streaming media. ;-)