Migrating IMAP email accounts to new hosting company

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I am moving a site from one hosting company to another.  The domain name
has email accounts with many IMAP folders in them.  Is there an easy
way to transfer the email accounts (including folders and email) to the
new server?

I can think of a complicated way to do it, by dragging and dropping with
an email account at another domain in my email client, but there must be a
better way.

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