To transfer emails from MAGIX Online World mailboxes to other hosting providers, please proceed as follows:

• Move the domain(s) (e.g. my-name.com) to your new desired hosting provider and set them up there again, so that new emails arrive there.
• To transfer old emails from MAGIX Online World to the new mailboxes, you can use free additional software such as "MailStore Home", which can also be used to create a private emails archive. 
• Download "MailStore Home" here: https://www.mailstore.com/en/products/mailstore-home.
• Set up a new archive from an IMAP account in the program following the instructions found here   https://help.mailstore.com/en/home/Archiving_Email#Archiving_from_IMAP.2C_POP3_or_Microsoft_Exchange_server.    

You can find the IMAP server to be used in the Control Center of MAGIX Online World under "Domain & e-mail" / "My email mailboxes".
Within each mailbox, there is an "Options" category with the subcategory "Email program settings".

• After archiving, follow the instructions provided by  "MailStore Home" to upload your emails to the new mailbox. As a rule, the export also takes place via IMAP. The instructions can be found here:  https://help.mailstore.com/en/home/Exporting_Email
• Please do not forget to reconfigure other devices (smartphone/tablet) and/or mail programs installed on the PC (e.g. Outlook, Windows Mail) for the new mailbox, so that you can continue to send and receive emails with these devices.
• Find additiona detailed support for "MailStore Home" here:

https://help.mailstore.com/en/home/MailStore_Home_Help