[git] How does origin/HEAD get set?

It is your setting as the owner of your local repo. Change it like this:

git remote set-head origin some_branch

And origin/HEAD will point to your branch instead of master. This would then apply to your repo only and not for others. By default, it will point to master, unless something else has been configured on the remote repo.

Manual entry for remote set-head provides some good information on this.

Edit: to emphasize: without you telling it to, the only way it would "move" would be a case like renaming the master branch, which I don't think is considered "organic". So, I would say organically it does not move.