I have a lot of those special folders, and it's been a pain in the wrist to scroll, tap, repeat. I keep almost all incoming email folderized, except for few that does not belong anywhere, which gets archived in a specific folder corresponding to a year. It's extremely easy to do this on Outlook or desktop apps, sadly no such a thing exists on iOS.
I tried Altamail, which was excellent, however their notifications do not work on iOS.
I guess the userbase of this forum is mainly leisure users, app addicts/social media, casual email communicators. Is there a forum where business professionals who get over 100 emails a day gather and discuss options to make use of their iphone? I gather most people just read the email and let it go, don't do much of a responding or replying back, unless it is a text message.
I just right click on a message or a group of messages and select a folder "on My Mac" and send it of to a named folder.
Sure it's hard if there is a lot of email in the 'Inbox'. But after you get it all saved and out of the Inbox the new ones get saved pretty quickly.
If you have no more email in the Inbox on OS X then run the command at the Apple Discussion post Why is Mail running slow. If you run the command (while mail is closed) it will speed up mail but archiving saved email (like Windows does if run).
I try. Most of the emails I get go straight to the trash anyway. Only a few are worth keeping. For work, there are a few actionable items I get, so they get flagged or sent to a folder for later.