One reason some emails are not showing (not downloaded) in Outlook could be the time setting determining how long Outlook has been set to keep email offline.

This article will guide you through how to change your Outlook settings.

Instructions may vary depending on the device and version of Outlook you are using.
If you require additional support for Outlook, we recommend reviewing the support resources provided directly from Microsoft:  https://support.microsoft.com/en-au/outlook 

1. Select File
   
   
   
   
2. Select the required email account
   
   
   
   
3. Select Account Settings, then select Account Name and Sync Settings
   
   
   
   
4. Change the Keep email offline for slider to the desired time period, then select Next
   
   
   
   
5. Select Done