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.
Your users can start using the new mail system immediately.
New mails are received in the the messaging system.
Old mails are migrated in a single pass.
This is probably the quickest, simplest and most cost effective solution.
Block the access to the old messaging system once you have switched your MX records.
Since there will be only one migration pass, if a user is still using the old system after the mailbox has been migrated, new items created in the old system will not appear in the new system.
Cloudiway migration benefits
Cloudiway migration can perform a one shot migration.
Mails older than a certain value can be migrated in the online mailbox, in the target archive or ignored.