For the last several years I've been involved mostly in tenant to tenant migration projects. These projects typically include migration of email, teams, and SharePoint. My focus is on email migration.
For migration, there are a number of third party tools such as MigrationWiz (Bit Titan) and On Demand Migration for Email (Quest ODME) that can be used. I don't know all of the details for all tools, but at least in the case of ODME, the migration is copies mailbox data, but doesn't synchronize it. So, if you migrate data from source to target and the user later deleted data from the source, it is not deleted in the target. Occasionally, this is annoying and causes confusion for users.
Microsoft has tenant-to-tenant mailbox moves in Preview. This type of move is a migration batch like moving mailboxes between on-premises Exchange and Exchange Online. In this case, you get true syncing which is a better quality move. It looks like a bit of a pain to setup and manage, but I expect the experience will improve over time.
The documentation claims that mailbox permissions like Full Access are maintained as long as users are part of the same migration batch. It does not mention inbox rules.
Another interesting item that's coming up is Microsoft 365 cross-tenant SMTP domain sharing. This allows an email domain to exist in two Microsoft 365 tenants simultaneously and will make it easier to manage incremental migrations between tenants when you want to maintain the same email address. The feature is currently in private preview, but is scheduled to be moved into public preview in June 2022.
You can monitor the development status at: