For cross-forest migration, you need have Exchange server in both forest.
You need to create forest trust before migration, then prepare mailbox for cross-forest migration. You could follow this blog to cross forest migration.
Please Note: Since the web site is not hosted by Microsoft, the link may change without notice. Microsoft does not guarantee the accuracy of this information.
About migrate mailbox, you could create migration mailbox one by one, you can also migration a part of mailboxes in a batch.
If you want to use old forest domain name in new forest, you need to remove the forest trust after migration, then add old domain as accepted domain in new forest. Then change MX record to the new forest.
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