Use the Embedded Twitch Javascript version.
Combined with JQuery, you can then do an onClick on another stream, Which does the following in pseudo:
mainplayer.pause();
animateMainOut(2); // Some kind of custom function that animates the currentplayer out
// ^ The 2, or whatever, should be some kind of index it moves the player to
// not sure what kind of lay-out you want to use so, may or may not use an argument
animateIn($(this)); // custom function animating the clicked object.
mainplayer = this.player;
mainplayer.play();