Merge branches without checking out branch

I have 3 branches.

     master [ Live Server]
      \
       stage [ Stage Server Where we test changes; merge commits ]
        \ 
         Dev [ Local Machine ]

I would like to downstream the changes to. Each of these branches are set to tracking each other.

Normally, to downstream the changes i do this:

git checkout stage && git merge master

Then i checkout dev and i do the same

git checkout dev && git merge stage

Then push them all: git push origin --all

Is there a way to downstream those changes without checking out into each branch?

I maybe using the wrong terminology. I'm not totally sure if i'm using upstream/downstream terminology correctly.