Internal moves need the same care as departures.
WHAT TO CHANGE
Role address forwarders: remove them from the ones they no longer handle, and add the person taking over
Shared mailbox access: remove and add accordingly
Any filters or rules they created in shared mailboxes Their signature, to reflect the new role
WHAT TO LEAVE
Their personal address, which follows them.
WHAT PEOPLE FORGET
Third-party services registered to their address on behalf of the business: analytics, suppliers, the domain registrar, the payment gateway. If those stay tied to a person who has moved on, the next problem is discovered at the worst moment.
Move business registrations to a role address, not another individual.
FOR CUSTOMER-FACING ROLES
Tell regular customers who now handles them. A forwarder is a technical fix; a short message is the relationship fix.
RECORD IT
Update your list of who holds what. The handover is only complete when the documentation matches reality.