We need to migrate a number of mailboxes for our users from our tenant to another tenant. We do not have access to a third party application to provide this. After some research I can see that this may be possible from on-premise to online, however I cannot see clear guidance on online to online. Could you please provide instruction / guidance on how best to migrate this content. My research also suggested that export and import of PST files for the users could be one means of achieving this. Again, could you please provide instruction / guidance on best practice to achieving this.
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.