If you’ve decided to move to another email provider it’s possible to take all of your old emails and folders with you. The easiest way I’ve found to do this is using the mail client Mozilla Thunderbird.
With Thunderbird installed sign into both your old and new emails accounts. This is provider dependent but in general if you are using a popular email service like Gmail, Yahoo, Outlook, etc. then Thunderbird can auto discover the SMTP endpoints. If you have two-factor authentication setup on your email account you may need to create an app password.
If you are unsure here are the instructions for a few popular services:
When you set up your old account make sure you set Thunderbird to download the entire email history not just the last few months.
Once you are signed in to both accounts you should see all of your emails and folders in the old account. You may want to wait for Thunderbird to finish downloading emails if necessary.
To move emails, simply select the inbox of your old mail account, use Ctrl + A to select all the emails, then drag them to the new inbox. You will also need to drag each of the folders from the old email account to the new one.
If you’d like to just move a couple of emails you can select them individually and drag them to the new email account.