It turns out this can be done using e-mail. In this case the client has a secure mail back-end (Exchange) and remote staff can use either Outlook or Outlook Web App (web mail) for a secure connection to Exchange.

The trick is to send a ZIP file as an attachment to each staff as internal mail (don't send to any external mail domain). Then the user can open it with Outlook or OWA and save the attachment to their local computer.

If Exchange is hosted you can delete the mails once staff have received their attachments so rogue hosting employees or three letter agencies have less chance of getting them.

I should have thought of this earlier but I've spent so much time with insecure mail systems that I rejected the idea without thinking it through.