What is the simplest way to move emails between two IMAP accounts?

From what I have seen, the simplest method for most people is using a standard email client that supports IMAP on both sides.

You add both accounts, let the folders sync, and then copy or drag emails from one mailbox to the other. It works reasonably well for smaller mailboxes and does not require advanced setup.

That said, it can get slow, may fail on large folders, and usually needs the client to stay open until everything finishes. Provider throttling can also interrupt the process.

I am curious how others handle this. Do you prefer client based copying, server side methods, or something else that has worked reliably for you?