Hi DGC,
Ideally the batch will sync all the new mail items of the synced mailbox between the on-premises server and Office 365. If the new mail items can’t be synced to Office 365, we can check if there are any error logs of the specific mailbox in the migration control panel of EAC (Exchange Admin Center). You can view the detailed error message by click “View Details” under the “Mailbox status” in the migration tab. After that, select the un-synced user and check the error message in the right panel.
Also since there are various causes for the issue, to solve the issue more quickly, I suggest you post a new thread with the error messages and details in our forum, thus one of our supports will focus on the issue and assist you in resolving the issue, thanks for your time.