You are correct. Muting a conversation archives it and any new emails that fall under the conversation (same subject) will skip the inbox and be archived.
I'm guessing you might be stuck on this next part though, as it seems to lessen the value of the feature itself.
Muting a conversation does not normally work if your email is in the To or Cc field. However, there is now a feature in Gmail Labs called Smart Mute that will keep a thread archived as long as your email address was not added to the To or Cc after you clicked Mute.
Alas, the "mute" feature only keeps incoming messages from putting that thread back into the inbox. (1) It doesn't keep the thread out of a label even if you use a label as a secondary inbox. (2) There's no equivalent of "mute" to mute a thread from another label. (3) Filters can't test if the incoming message is going into a muted thread.
Archive or mute Gmail messages says:
When you mute a message: Any replies stay out of your inbox. You can search for the conversation if you want to find it again.
And in the Help Forum:
if you mute a conversation from a mailing list new emails in that thread wont appear in the inbox but will be archived in all mail until you unmute it or one comes addresses directly to you
Filter criteria seem to apply to incoming messages before filter actions. They apply after spam classification even though the UI warns otherwise. So this does work:
Matches: in:spam (from:("christianmingle" OR "DeVry University" OR "Fidelity Life Insurance" OR "Walmart Points"))
Do this: Delete it
If -is:mute worked in a filter (i.e. if the test applied to the new message's thread instead of to the new message itself, which isn't yet in a muted thread), you could use one filter to skip the inbox and a second filter with the same criteria plus -is:mute to apply the label.
Or similarly, if the filter criteria could test labels on the new message's thread, you could move your undesired threads to a different label muted as a hack way to mute them, then use a pair of filters with -label:muted as part of the second filter's criteria.
But Gmail doesn't currently work that way. You can use the Send Feedback feature (in the gear menu) to request one of these tweaks.
As a desperate way to handle an occasional annoying thread, you could add a filter just for that thread which just removes it from the Inbox and the alternate lable.
More ideas: Write a Google Apps Script, or a Greasemonkey script, or a Chrome extension to extend Gmail.
Best Answer
In the Inbox web UI there is no menu option for Mute, but you can use the M key. Sorry I didn't see the "mobile" part of the question before I answered.
There is no way to Mute in the mobile app AFAIK at this time.
Hitting ? will display the keyboard shortcuts and allow you to enable / disable them.