Avast Security was causing Mac Mail (Version 9.3, Mac OS 10.11) to continually re-download all mail files. I eliminated the problem by adding ~/Library/Mail/ to the Excluded Paths in File Shield under Preferences of Avast Security.
would you mind to send us your Support package? https://support.avast.com/en-ww/article/Submit-Mac-Security-support-file
It may help us to nail the root cause.