I can't speak for Purely, but every one I've tried has had an "import from [previous provider]" feature because that's basically just "pull from imap".
Otherwise: for ~20 years now I've been able to just attach two imap accounts in thunderbird, and drag to move/copy everything from one to the other, and that'll just chug away until it's done. I've never had an issue doing that.
Otherwise: for ~20 years now I've been able to just attach two imap accounts in thunderbird, and drag to move/copy everything from one to the other, and that'll just chug away until it's done. I've never had an issue doing that.