When using Microsoft Exchange ActiveSync (EAS) with mailboxes that are over 5 GB in size, you may experience some unexpected behavior, including sync key mismatches, general slowness or an overall poor user experience. This is typically seen when connecting to newer versions of Outlook for Windows (Outlook 2013+); however, these issues could impact mobile sync as well. (Those syncing to Outlook365 or Outlook.com should not experience these issues as these services utilize MAPI over HTTP, which is a completely different protocol than used for EAS.)
While we do recommend that those with large mailboxes implement IMAP for syncing, we realize that this is not possible for every environment. One way to resolve these EAS syncing mishaps is to limit the timeframe in which emails are synced from the server to no longer than 1 month. This limit can be put in place by adjusting the sync settings on both mobile devices and Outlook alike. (In Outlook, adjust the "Mail to keep offline" setting. iPhones: "Mail Days to Sync". Android devices: "Period to sync Email".) Keep in mind that when using mobile devices, email over a month old will not be stored; however, you can use the search functionality to return results for older items on the server when necessary.
By making this small adjustment, the impact on the server and EAS should be less taxing, which should alleviate the unexpected syncing behavior.