That process works if your using IMAP protocols for your email programs. IMAP keeps email on their servers and doesn't let the user actually download it to their PCs for storage. That's the way I prefer to do things. I don't want my emails to stay anywhere else but here locally. That's also why I never use "the Cloud" for data storage. Again, everything is kept here locally, with redundant backups kept safe in Faraday containers.
Call me "old fashioned" but in my view, my data is better kept secure with me, not with anyone else.