Looking for a solution for the following.

Clients have mailboxes under their own domain name i.e. user@domain.com.

The client only wants to use IMAP as they use several devices with the same user/domain combination as they want all their devices to have access to the same folders/emails especially the sent items.

The issue is I dont have unlimited storage space so am looking to see if there is a way in which we can get all the client emails and folder structure to be backed up to one of their Gmail accounts.

i.e. user@domain.com ---> user@gmail.com

In so that they will always have a complete copy of the mailbox saved in their gmail account.

Hoping someone is already doing this and to hear the method they are using.

Thanks