With Horde, you can download the emails in a mailbox to your computer. For this, after logging in Horde web application in DYNAMIC mode, please right click on the mailbox you want to download and select Export
In the popup window, select "Download into a MBOX file" and click OK.
If you use Horde web application in BASIC mode, please use the Folders link, right below the NEW MESSAGE button, once you select the folder to be downloaded, click "Choose Action" menu and click Download.