[AC Question]

I'm using the stock email service on a Samsung A3 connected to my 365 account.

Its set to retain 7 days, but I often have to search back several months. This works fine when connected by wifi but fails with "no response, try later" when connected by mobile data.

I also find that whilst I do receive some new emails when out and about on mobile data, there seems to always be a bunch of new ones arriving when I get back to a wifi connection.

[ManiacJoe]

In the app settings, check the wifi vs data settings for sync and download limitations.
In the app, check the account settings for similar limitations.

The default apps tend to be limited in their abilities and options.
You may need to update to a more advanced, third-party, email app.