How can I download messages from OWA to Outlook or Outlook Express?
You can't. (Well, you probably can... it's only software after all, but
there is no current mechanism to do so)
OWA doesn't 'download' anything. It is a web page rendering of your
mailbox. Where is this 'set to download messages from the Exchange server'?
I can think of two possibilities:
1) You have accessed the same mailbox with Outlook where the delivery is set
to a local .pst file, and therefor older content is not in your mailbox
or
2) You have your OWA view set to unread messages only or some other view.
This is a drop down option by the folder name at the top of an OWA view.