I get that sometimes, too, where iOS Mail app is quick to get new emails but doesn't really change its status to show an email was moved or deleted. I will get alerts at the exact same time on my phone as Outlook 2016 so its certainly getting pushed to very fast. Mail app on Yosemite is slow to show new emails. I can have both Mail app on Yosemite and Outlook 2016 both open, and Mail app will take minutes to show a new email that Outlook 2016 (and my phone) already alerted to.
I am not sure if this is a client issue or Office 365 but I suspect a client issue due to the inconsistency between clients. If I mark an email as read on my phone Outlook 2016 drops the alert bubble on the dock icon within seconds while OS X Mail app will still have the alert bubble for minutes. If I mark an email as read in Outlook 2016 both OS X and iOS Mail app will still show an alert bubble.
Thanks for that feedback, much appreciated!
With different private Microsoft Exchange server at work, as well as connecting to Gmail using Exchange Activesync (paid through google apps with a private domain), I get full instant syncing of everything - not just the mail being pushed to the devices, but also read status / deletes. For example, if I receive a new mail item, I see it on my Mac, and the lock screens on both iPhone and iPad. As soon as I delete it or mark it as read, it disappears from the lock screen of my iOS devices quite instantly. I should note though that in the Mail app on Mac, I use IMAP to connect to Gmail, and it pushes new items and changes quite fast (using IMAP IDLE, which iOS doesn't support).
Office365 is doing a great job of pushing items instantly to iOS devices. And as soon as I mark as read, delete, move to a folder, or flag it, I see that change show up instantly in OWA (I've been keeping OWA open for reference). I do notice, as you said that it does take a few minutes for new items and changes to get pushed to the Mac Mail app. Though I'm quite happy I can flag items on the go that I need to follow up on and see them in the Mail app on Mac. But the one thing that I find hard to live without is read/delete/move/flag statuses not being pushed to iOS devices after being used to that for 6-7 years now. When I'm at my desk going through several emails on the Mac, and then grab my iPhone and see the lock screen filled with them, it drives me nuts. The only way to get the iOS device to update those changes is to unlock it, open the Mail app and let it refresh, which is unfortunate. I'm going to keep using it with this one domain for a while and hope that the behaviour changes.