In the navigation menu at left click on your general "mail" folder (which should contain all your mail sub-folders)
Next, in the menu bar at top, click on "Folders."
Now select the folder you wish to export by checking its box.
Next, select "Download Folder(s)" from the "Choose Action:" drop-down box.
You will be given an .mbox file to download. Store it in a location you'll remember.
Importing Your Emails Into Horde
Click "Folders" in the menu bar.
Select "Create Folder" from the "Choose Action:" drop-down box and give your new folder a name (preferrably the same name as the one you just exported from).
Now select this folder by checking its box.
This time, select "Import Messages" from the "Choose Action:" drop-down box.
Next you'll browse to the download location you used above and select your .mbox file to import and click "Import."
Your messages should automatically be imported into your newly created folder.
Double check the new folder and make sure the messages are there and that they are intact.