I'm just switching my hosts and I want to move the emails from my accounts too. In order to have my current emails on the new host I want to download/export them and to import at the other host. In order to do this I use one of the webmail (squirrelmail, roundcube, horde) clients available on my current host.
The problem is that except for roundcube, I don't see any download/export option available. And in roundcube I can only select one email at a time and download it as eml.
My question is how do I export/download all the emails from one account and import them at the new host? I know this is possible because I remember doing this some time ago using squirrelmail, but I can't find anything related to this now.