By using cPanel mail service you can back up (export) your emails using both web interfaces - Roundcube or Horde. Built-in restore (import) option is available in both Roundcube and Horde webmail now.
Follow the steps in this article to access cPanel webmail:
In order to back up (export) emails to your local computer, select the required email(s) and click on More icon in the bar above:
Select Download (.eml) in dropdown list option and specify the location to save the email(s) to:
NOTE: .eml files are supported by most of the email services and can be easily restored later on different servers. Note that, Roundcube does not provide convenient functionality to download many emails at once, so if you want to back up the whole folder or account, we recommend using Horde instead.
To import the emails using Roundcube webmail, click on Gear icon >choose Import messages:
After this choose the backup file you wish to be imported and click Upload:
When the process will be finished, you will get a notification about successful import:
The single mail backing up the process in Horde is just the same as that in Roundcube - choose the email you want to back up and select Save option in the dropdown menu. Email will be also saved in .eml format in this case.
To save all emails in the folder (Inbox, Sent, etc.) right-click on this folder and select Export option.
Then you should select the format of the backup file - MBOX or zipped MBOX. If you just want to store emails, we advise choosing the zipped option as it will be smaller in size.
The mbox file can be used for emails transfer to another cPanel account with Horde available (via Import option).
For importing emails in Horde right-click on the folder you wish to upload the emails to and choose Import.
Then select the backup file in Mbox or .eml formats and click OK.
Now your emails are imported to the email account. You should get a pop-up notification the emails have been uploaded from the backup file to the account.