If you want to go from:
 x--x--x--x--x    (origin/master)
        \
         Y--y--y  (origin/abranch)
to a local orphan branch:
Y--y--y (abranch)
You need to:
Depending on your case, you might find Y (the first commit from origin/abranch not in origin/master) with "how to find first commit of specific branch":
git log origin/master..origin/abranch --oneline | tail -1
If origin/abranch has a single merge base as shown above, then use git merge-base to find Y^ (parent commit of Y, since cherry-pick does not include the first commit itself):
git cherry-pick $(git merge-base origin/master origin/abranch)..origin/abranch
Or does it matter what branch another branch is created from, if I want to add those changes back to a particular branch?
A branch is just a pointer and help referencing all commits accessible from its HEAD. If a branch is spawned from another, that means all of the other branch commits are part of the new branch as well.
See "Using branches" and "Git Branching - Branches in a Nutshell".
It is best to merge back a branch in its the branch it came from (not exactly a "parent" branch as there is no concept of ancestors for branches)
  x--x--x--x--x--M (master)
   \            /
    Y--y------y    (abranch)
If you don't, that means oyu are cherry-pick or rebasing --onto, which is more complex and risk duplicating commits.
That would go  from:
 x--x--x--x--x         (master)
        \
         p--p--p       (anotherbranch)
             \
              Y--y--y  (abranch)
To (with git rebase --onto master $(git merge-base anotherbranch abranch) abranch) :
               Y'--y'--y' (abranch)
              /
 x--x--x--x--x            (master)
        \
         p--p--p          (anotherbranch)
Then you can fast-forward abranch to master.
To experiment some more with branches, see "Learn Git Branching".