Only until Microsoft Outlook has synchronized with the mailbox, that the most up to date data will be displayed to the user.
Thus, your Inbox will appear to have a set of missing emails. The problem is that IOS Apple Mail hasn't "fetched" those gap emails yet and therefore is not showing them in your other email clients on the face of the planet (including Apple OSX Mail) fetch from your mail server by latest arrivals first instead of most recently copied or moved.
Apparent gaps will be magically healed as Mail app caches the entire Inbox.
This seemingly endless trip to the bottom of your Inbox is tedious, but it really does set things straight.
Note: In case you have not yet installed any of these updates, do not go for them.
However, if you have already done installing any of those, you must go on with this post to resolve the issue.