diff --git a/.github/workflows/triage-issues.yml b/.github/workflows/triage-issues.yml
index 29e08979e6..8273db9e17 100644
--- a/.github/workflows/triage-issues.yml
+++ b/.github/workflows/triage-issues.yml
@@ -5,6 +5,8 @@ on:
   push:
     paths:
       - .github/workflows/triage-issues.yml
+    branches-ignore:
+      - dependabot/**
   schedule:
     # Once every day at midnight UTC
     - cron: "0 0 * * *"
@@ -22,7 +24,7 @@ jobs:
     runs-on: ubuntu-latest
     steps:
       - name: Mark/Close Stale Issues and Pull Requests
-        uses: actions/stale@v3
+        uses: actions/stale@v3.0.18
         with:
           repo-token: ${{ secrets.GITHUB_TOKEN }}
           days-before-stale: 21
@@ -36,6 +38,29 @@ jobs:
           exempt-issue-labels: "gsoc-outreachy,help wanted,in progress"
           exempt-pr-labels: "gsoc-outreachy,help wanted,in progress"
 
+  bump-pr-stale:
+    if: >
+      startsWith(github.repository, 'Homebrew/') && (
+        github.event_name != 'issue_comment' || (
+          contains(github.event.issue.labels.*.name, 'stale') ||
+          contains(github.event.pull_request.labels.*.name, 'stale')
+        )
+      )
+    runs-on: ubuntu-latest
+    steps:
+      - name: Mark/Close Stale `bump-formula-pr` and `bump-cask-pr` Pull Requests
+        uses: actions/stale@v3.0.18
+        with:
+          repo-token: ${{ secrets.GITHUB_TOKEN }}
+          days-before-stale: 2
+          days-before-close: 1
+          stale-pr-message: >
+            This pull request has been automatically marked as stale because it has not had
+            recent activity. It will be closed if no further activity occurs. To keep this
+            pull request open, add a `help wanted` or `in progress` label.
+          exempt-pr-labels: "help wanted,in progress"
+          any-of-labels: "bump-formula-pr,bump-cask-pr"
+
   lock-threads:
     if: startsWith(github.repository, 'Homebrew/') && github.event_name != 'issue_comment'
     runs-on: ubuntu-latest