I've had a response from Apple engineers:
The customer is reporting that POP emails are not opening after updating to iOS 9. iOS Mail caches the email, and then that cache may clear when space is needed. It will then download them again from the server.
This issue usually starts if one of the devices/computers that was setup with this email account has been configured to remove a copy of emails on the server. If this is the case, the customer will need to check the settings of all the devices and computers that are running this email account.
iOS leaves messages on the POP3 server by default. iOS never stores email permanently, whether POP3 or IMAP. If they are removed from the server by another device then this issue can occur.
A test that you may want to do to confirm this is to have the customer produce the error by opening one affected mail on the iOS device, and then locate that specific email on his webmail. If not found, then the device is behaving as expected. If found, please try to forward it to the same mailbox and see if the same issue occurs.
I look forward to your follow-up on this issue with the results of checking data on the server.