One alternative to the migration path mentioned in other articles is to start with empty mailboxes and migrate just the last X days of mails (for example, 30 days of mails). Once the cut-over is complete, you can take your time to migrate the older mails either to online mailboxes or directly to archives.
What is a Pre-staging migration? A pre-staging migration is a multipass migration where you will migrate most of the mailbox items before the final cut-over.
What is coexistence?
Big bang (one shot) migration This scenario is the simplest to implement. After you have switched your MX records to point to the new system, you start mailbox migration.
There are several ways to migrate your messaging system. This article gives a short introduction to the different options.