You certainly should not change email providers like you change your underwear. However, sometimes change is necessary. If you decide to change your email host, you must notify the email host you’re leaving. Here’s why:
First and foremost, you do not want unused email boxes just sitting on a server. When things go unchecked, they become vulnerable to hackers. Should you decide to switch email providers, it is wise to close down any and all old email accounts to protect yourself and those in your contacts list from malicious emails, hacking, and more.
It is important to note that email providers do not check email accounts for use. If you’ve abandoned an email account, your host will not know unless you speak up.

Local governments and many other public entities must comply with Right to Know laws. Internal emails are considred public records and must be available if requested. Should you switch emails and just delete the account, all of the old emails go with the account. They cannot be recovered. Before switching email hosts, it is wise to get a backup of all emails sent to comply with federal, local, and state regulations.
If you’ve already switched email providers and realize now that you have old accounts sitting out there, you should take immediate action. Talk to your host about getting a backup or archive of the emails. Then shut it down so that hackers do not gain access to the account.
It is important to note that an ISP will NOT delete old email accounts for you. It is always the owner’s responsibility to comply with any laws and to remove and/or delete emails, email boxes and email accounts.