How do I move a search-filtered set of emails into another folder?

How do I move a search-filtered set of emails into another folder?

I'm trying to clean up my gmail inbox which has become swamped. My first
step is to unclutter it by breaking it up into folders of related mail.
That will let me deal with incoming mail appropriately and attack the
older mail in chunks as I can get to it.
My problem is detailed below, but is, in essence, "How do I move a
search-filtered set of emails into another folder?" Not to the archive.
Not merely labeling them (which drag-and-drop also does). Moving them from
the Inbox to another specific folder in my own folder hierarchy - and not
all emails in my inbox, just the specified ones (as opposed to this post:
How to move all (not just showing) mails in gmail to another folder?).
Here are the details (aka, what I already know how to do):
It is easy to check a number of emails on a given page. If one wants to
select specific emails (say, from a given sender in this example) out of
the Inbox, one can use a search filter.
One can also check the emails selected by the filter in a method similar
to just checking them in the Inbox.
The problem is that the Move button now changes from a picture of a folder
with the dropdown to choose which folder, into a button with the text
"Move to Inbox". The messages are already in my Inbox. I'm trying to move
them out into another folder. That's what most people would want to do,
most of the time. Yet, that's the one functionality that has been taken
away.
So how do I move those emails into a folder?
Being able to do it for more than a page's worth of emails would also be
handy. FWIW, the Qs and As posted so far on this site (and others, like
here:
http://www.justanswer.com/computer/4e06z-gmail-how-move-large-number-emails-folder.html)
don't answer this question, at least not that I have found.
If there is another article, or version, or I'm missing something obvious,
that's fine. I'm not proud. Tell me it's simple. So much the better.
I just want an answer, as not having one is the reason my Inbox has become
unmanageable to begin with.
My setup: Win 7 Chrome