diff --git a/.github/workflows/ci_cron_weekly.yml b/.github/workflows/ci_cron_weekly.yml
index e3dfbd4..15d0703 100644
--- a/.github/workflows/ci_cron_weekly.yml
+++ b/.github/workflows/ci_cron_weekly.yml
@@ -1,27 +1,60 @@
-# GitHub Actions workflow that runs on a cron schedule.
-
-name: Cron Scheduled CI Tests
+name: Weekly cron
 
 on:
+  pull_request:
+    # We also want this workflow triggered if the 'Extra CI' label is added
+    # or present when PR is updated
+    types:
+      - synchronize
+      - labeled
   schedule:
-    # run at 6am UTC on Mondays
+    # run every Monday at 6am UTC
     - cron: '0 6 * * 1'
 
+concurrency:
+  group: ${{ github.workflow }}-${{ github.ref }}
+  cancel-in-progress: true
+
 jobs:
-  # Testing links in documents is a good example of something to run on a schedule
-  # to catch links that stop working for some reason.
-  doc_test:
-    runs-on: ubuntu-latest
+  tests:
+    name: ${{ matrix.name }}
+    runs-on: ${{ matrix.os }}
+    if: (github.repository == 'astropy/specutils' && (github.event_name == 'schedule' || contains(github.event.pull_request.labels.*.name, 'Extra CI')))
+    strategy:
+      fail-fast: false
+      matrix:
+        include:
+
+          # We do not use remote data here, since
+          # that gives too many false positives due to URL timeouts.
+          - name: Python 3.11 with pre-release version of key dependencies
+            os: ubuntu-latest
+            python: '3.11'
+            toxenv: py311-test-predeps
+
+          - name: Documentation link check
+            os: ubuntu-latest
+            python: '3.10'
+            toxenv: linkcheck
+
     steps:
-    - uses: actions/checkout@v3
-    - name: Set up Python to build docs with sphinx
+    - name: Checkout code
+      uses: actions/checkout@v3
+      with:
+        fetch-depth: 0
+    - name: Set up Python
       uses: actions/setup-python@v4
       with:
-        python-version: 3.8
-    - name: Install base dependencies
-      run: |
-        python -m pip install --upgrade pip
-        python -m pip install tox
-    - name: Check links in docs using tox
+        python-version: ${{ matrix.python }}
+    - name: Install language-pack-de and tzdata
+      if: ${{ matrix.os == 'ubuntu-latest' }}
       run: |
-        tox -e linkcheck
+        sudo apt-get update
+        sudo apt-get install language-pack-de tzdata
+    - name: Install graphviz
+      if: ${{ matrix.toxenv == 'linkcheck' }}
+      run: sudo apt-get install graphviz
+    - name: Install Python dependencies
+      run: python -m pip install --upgrade tox
+    - name: Run tests
+      run: tox ${{ matrix.toxargs}} -e ${{ matrix.toxenv}} -- ${{ matrix.toxposargs}}
diff --git a/.github/workflows/ci_workflows.yml b/.github/workflows/ci_workflows.yml
index 6d3db46..cc23855 100644
--- a/.github/workflows/ci_workflows.yml
+++ b/.github/workflows/ci_workflows.yml
@@ -25,6 +25,11 @@ jobs:
             python: 3.x
             toxenv: codestyle
 
+          - name: Code style checks
+            os: ubuntu-latest
+            python: 3.x
+            toxenv: build_docs
+
           - name: Python 3.11 with astropy data and coverage
             os: ubuntu-latest
             python: '3.11'