Sorry, I got a bit offended when you said I missed the point, as if I didn't understand the problem, because I was looking for he same solution as everyone here. I have a galaxy s5, which I just got, so it's kitkat. I have another update pending as of this morning, and I updated it once just after activation on Wednesday.
I have moved two emails from the same inbox on a pop account (comcast) to the same folder that is named "archive" in my icloud account, which is imap. I am using the email app that comes on the phone, and all the accounts except one was auto setup by the app. The one I had to fight with and manually install is not one of the accounts to or from which I'm moving emails.
I don't like exchange accounts or use Outlook because I tend to clear things off my phone to unclutter the inbox, but I don't want them permanently moved or deleted. I will sort them again from my laptop if I need to print one or keep it in the inbox to remember to bill for it. And I don't download through outlook so they stay on the server and I can see the same stuff on either the home or office computer.
In the email, when I click the menu dots, it opens this:
Then when I select "move", it opens a box that lists all five of my email accounts, and they are expandable to show their folders. I have one account that was treated as an exchange account on iPhone that is no longer treated that way on the galaxy (I didn't set it up like that on either platform but when I deleted or moved emails in thar account, from my phone, they were moved from the server). The folders that I created on that particular account do not all show up, and didn't on my iPhone either, and I have a file named "saved" on it as well, where I would keep important emails, but couldn't from that phone, and would instead put them in the archive folder of my icloud email account. But that is a free jimdo account which might explain that issue. It's a problem to set up on both phones, so there's something different about it.
When i select "move," the menu looks like this (just my icloud folders in this shot so it doesn't show my email addresses):
When I select the folder, the email is moved. It takes a few minutes for it to show up, but eventually, it works.
I don't know how or why I have this option. If I select "save email" instead of "move," it goes to the folder in the combined inbox named "saved emails," which will also work for my purposes, but not every one else's. I'm only keeping emails in this folder on my phone that I might need when I can't get to my computer to get them: car rental and hotel confirmstions, emailed coupons, etc.
I hope someone else can duplicate this. Again, it's from within the email, so not as convenient if there's more than one or two, but hopefully doable by someone besides me.