To improve the speed of migration and reduce your organization’s bandwidth constraints for Exchange Online, consider the following:

• Reduce mailbox sizes and always use cached mode
  • Smaller mailbox size improves migration speed. Cached Exchange Mode enables a better experience when you use an Exchange account. In this mode, a copy of your mailbox is saved on your computer. This copy provides quick access to your data, and it is frequently updated with the server that runs Microsoft Exchange.
• Use the mailbox move capabilities with an Exchange hybrid deployment:
  • With an Exchange hybrid deployment, offline mail (in the form of .OST files) does not require re-download when migrating to Exchange Online. This significantly reduces your download bandwidth requirements. A hybrid deployment provides the seamless look and feel of a single Exchange organization between an on-premises Exchange organization and Exchange Online in Microsoft Office 365. In addition, a hybrid deployment can serve as an intermediate step to moving completely to an Exchange Online organization.
• Schedule mailbox moves to occur during periods of low Internet traffic and low on-premises Exchange use:
  • When scheduling moves, migration requests are submitted to the mailbox replication proxy and may not take place immediately.