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This may not just be Office 365 but is so in our case. We are migrating 12 accounts from Office 365 to SmarterMail 16.
Whether we choose to copy or move it says it cannot be done when using ActiveSync. If we delete the ActiveSync and change to IMAP we can copy and move from Office 365 to SmarterMail 16. We are not trying to copy under the Inbox as we've read that doesn't work but anywhere else is fine. We are using Outlook 2016 and SmarterMail 16 (both the latest releases out there).