If you don’t want to use the standard Microsoft tools for migrating Mailboxes to or from Exchange Online you have to use a 3rd party solution. One online solution is MigrationWiz from BitTitan. There are a number of advantages of using MigrationWiz over the standard Microsoft tools, especially when using a Cutover Exchange Migration (CEM) or a Staged Exchange Migration (SEM). A Cutover Exchange Migration is basically a big bang scenario where you switch all services at the same time and gradually migrate all the Mailbox content. There’s no way to schedule anything here and this is much better when using MigrationWiz.
MigrationWiz is using Exchange Web Services to login to your on-premises Mailbox and to your online Mailbox to move all the content between the two platforms.