What happens between Sauron and Saruman after the battle at Dol Guldur?

After the battle at Dol Guldur when Sauron is bansihed back to Mordor (I think it's Mordor), Saruman tells Elrond

Leave Sauron to me

What happens next?

Does Saruman travel to Mordor with the intention of finishing Sauron off, or was he always planning on joining him?