Migrate from one Office 365 Tenant to another

Best practice would be to use a third-party tool, as they can cover the entire process and make it a lot easier on both admins and end users. If you don't want to use third-party, you'll have to do it via manual pst export/import indeed, as Microsoft doesn't offer any tools for this scenario, apart for generic guidance. While this might work for smaller shops (lets say up to 100 users or so), it quickly becomes unmanageable as the number of mailboxes grows.