"You can now earn bonus reputation for your first dungeon and scenario of the day. You can select which reputation you choose to champion by selecting it from the reputation panel on the character screen. When you queue for a dungeon or scenario, the UI will remind you which reputation you are championing and allow you to change the reputation from there."
Will this work for Kirin Tor Offensive and Sunreaver Onslaught?
Can't find any official sources on it, and it may be obvious, but I'm sure it doesn't work with Shado-Pan Assault, either. As it is the new "raid" rep. You can get SPA rep in LFR though, so that's something.
"Join the Shado-Pan Assault in their singular drive to see Lei Shen, The Thunder King, defeated once and for all, and gain access to impressive Valor reward items. Reputation with this faction can be earned only within the Throne of Thunder raid dungeon."