diff --git a/.github/workflows/markdown-links.yml b/.github/workflows/markdown-links.yml
index 6aef1baf8..d9de1359f 100644
--- a/.github/workflows/markdown-links.yml
+++ b/.github/workflows/markdown-links.yml
@@ -1,23 +1,20 @@
 name: Check Markdown links
 
 on:
-  push:
-    branches:
-      - main
-  pull_request:
-    branches: [main]
+  schedule:
+    # 2am UTC daily
+    - cron: '0 2 * * *'
 
 jobs:
   markdown-link-check:
+    strategy:
+      matrix:
+        branch: ['main', 'v0.37.x', 'v0.34.x']
     runs-on: ubuntu-latest
     steps:
       - uses: actions/checkout@v3
-      - uses: technote-space/get-diff-action@v6
         with:
-          PATTERNS: |
-            **/**.md
-      - uses: creachadair/github-action-markdown-link-check@master
+          ref: ${{ matrix.branch }}
+      - uses: informalsystems/github-action-markdown-link-check@main
         with:
-          check-modified-files-only: 'yes'
           config-file: '.md-link-check.json'
-        if: env.GIT_DIFF
diff --git a/.md-link-check.json b/.md-link-check.json
index 6f47fa2c9..68f3c1482 100644
--- a/.md-link-check.json
+++ b/.md-link-check.json
@@ -2,5 +2,16 @@
     "retryOn429": true,
     "retryCount": 5,
     "fallbackRetryDelay": "30s",
-    "aliveStatusCodes": [200, 206, 503]
+    "aliveStatusCodes": [200, 206, 503],
+    "httpHeaders": [
+        {
+            "urls": [
+                "https://docs.github.com/",
+                "https://help.github.com/"
+            ],
+            "headers": {
+                "Accept-Encoding": "zstd, br, gzip, deflate"
+            }
+        }
+    ]
 }