release(patch): use the makefile defined in the release branch (#12483)

.github/workflows/run-patch-release.yml

@@ -24,6 +24,7 @@ jobs:
   prepare:
     runs-on: ubuntu-latest
     outputs:
+      release-branch: ${{ steps.prepare.outputs.release-branch }}
       release-version: ${{ steps.prepare.outputs.release-version }}
       slack-thread: ${{ steps.prepare.outputs.slack-thread }}
     steps:
@@ -42,6 +43,7 @@ jobs:
     runs-on: ubuntu-latest
     needs: [ prepare ]
     env:
+      RELEASE_BRANCH: ${{ needs.prepare.outputs.release-branch }}
       RELEASE_VERSION: ${{ needs.prepare.outputs.release-version }}
     permissions:
       contents: write
@@ -50,6 +52,8 @@ jobs:
         with:
           # 0 indicates all history for all branches and tags.
           fetch-depth: 0
+          # Use the makefile in the given release branch.
+          ref: ${{ env.RELEASE_BRANCH }}
 
       # Required to use a service account, otherwise PRs created by
       # GitHub bot won't trigger any CI builds.