Cross-forest permissions have long been one of the more difficult aspects of an Exchange Hybrid migration. Ensuring mailbox permissions are preserved can be a major undertaking.
Exchange Hybrid migrations did not support any sort of cross-forest mailbox permissions at first. After a couple of years Microsoft made it possible for full access permissions to work cross-premises.
Recently Microsoft has improved the cross-forest permissions story for Exchange Hybrid migrations. In this blog post, I’ll give you the information you need to set the different permissions in your Exchange Hybrid deployment.