When a faction is down to 1 province, make sure you bring enough archers (to shoot the faction leader since he can't be ransomed anymore).

Also, have a Steppe Cav (a must in an army; or any fast horse will do). Have it in reserve (don't use them in battle) about 2 1/2 rows deep but put it in wedge after that. Pin the AI leader with a spears and shoot him with archers. If he tries to escape, spring your steppe cavalry(should be close to the fight), he's going to be caught easy (but don't try to follow him with the other infantries, that will only slow down the fast horse).

I always catch the faction leader or important generals this way. Try to divide the AI's provinces so that you can ransom the faction leader (or heirs) multiple times till he is broke. Leaving a faction in an island is good too, so you can consolidate on the mainland. Most of the time a faction left on an island will be broke.

If the faction leader dies, then you got rebels that you can bribe or storm (depending on the situation).