Have you managed to transition off and gracefully deprecate old email addresses? How?
Similarly, get all your important contacts to update to the new address.
Set up forwarding from the old e-mail to the new one, if you can.
Stop sending anything from the old one.
I told everyone I know that I'm using a new email address, I set up forwarding, I only send using the new email address, and I have an auto-respond message that I'm not using that email address and to send to the new one instead, and nonetheless people still send to it.
Also, for security reasons I'd recommend to not close the old email address, because if you happen to forget an account there, there is some risk that someone nefarious could grab the old email address and have password resets sent there.