The easiest way for you to ensure your email is backed up is to keep your email account set up in your email application.
Add the email account hosted by us, as an additional account and keep the old account as an archive.
We do not recommend migrating your old emails to your new email account as this can fill the storage and cause sync issues.
If you view your emails online via Gmail or any other webmail, you will need to add your email account to an email application such as Microsoft Outlook, Mac Mail or Thunderbird, so that emails are downloaded and stored on your device. Your current email provider will be able to supply you with the relevant email account details to do this. If you can see your emails while offline then they are stored to the device.
In addition to this, we would recommend that you back up your emails to an external hard drive just in case anything was to happen to your device.
Here are some relevant links: