When you put the EE's in with the BO's, did you just place them in the coop? If so, there's one of your problems.
It's highly advised to quarantine new chickens away from your existing flock for around a month to prevent any disease/parasites/infection the newcomers may be carrying from spreading through your whole flock.
After the quarantine period is over, you can them put the newcomers in separate cage/crate in with the existing birds so they can get used to one another.
I know when I was introducing my Welsummers, my Araucana was fluffing up her breast feathers and striking at them through the cage wire, the cheeky beggar.
After a couple of days with each other, letting the new chickens out of the cage with the existing birds at night is a popular way to do it. The existing birds kind of wake up the next day, look at the new ones with a "oh, where have you been all this time?" look on their faces.
Pecking and going after each other is all part of normal behaviour as they try to re-establish a pecking order.
It's important to have enough space in the coop/run where the attackees can run away from the attackers. I don't have much issues with squabbles as I have only 9 chickens in a huge coop/run set up that could probably house 20-25+ birds.
At this stage, your EE's are at the bottom of the pecking order with the bantams. I don't know whether they will try to challenge them in their pecking order or not as I don't keep bantams.
This is all just from my experience - others may have more information on what you should do, but this is what has worked for me.