This is the exact same issue that was raised in JENKINS-20427, however it appears to not have been fixed.

Create a job with the "this build is parameterized" selected.
Input parameter into "branches to build" section for Git source code
(Note: tried /*${BRANCH} and simple ${BRANCH}

Any time any other branch in the GIT repo is pushed to, this job will kick off with that branch.

The intended use is for an ajax call to post into the build kicking it off with a branch of the developers choosing.