diff --git a/build.sh b/build.sh
index 6fefdeb4..c9a5e30c 100755
--- a/build.sh
+++ b/build.sh
@@ -76,6 +76,8 @@ docker buildx build \
 	--rm -t quay.io/pypa/${POLICY}_${PLATFORM}:${COMMIT_SHA} \
 	-f docker/Dockerfile docker/
 
+docker run --rm -v $(pwd)/tests:/tests:ro quay.io/pypa/${POLICY}_${PLATFORM}:${COMMIT_SHA} /tests/run_tests.sh
+
 if [ -d $(pwd)/.buildx-cache-${POLICY}_${PLATFORM} ]; then
 	rm -rf $(pwd)/.buildx-cache-${POLICY}_${PLATFORM}
 fi
diff --git a/docker/Dockerfile b/docker/Dockerfile
index fe112aa0..4949b154 100644
--- a/docker/Dockerfile
+++ b/docker/Dockerfile
@@ -160,12 +160,7 @@ COPY --from=build_cmake /manylinux-rootfs /
 COPY --from=build_swig /manylinux-rootfs /
 COPY --from=build_cpython /manylinux-rootfs /
 COPY --from=all_cpython /opt/_internal /opt/_internal/
-COPY build_scripts/finalize.sh /build_scripts/finalize.sh
-COPY build_scripts/python-tag-abi-tag.py /build_scripts/python-tag-abi-tag.py
-COPY build_scripts/ssl-check.py /build_scripts/ssl-check.py
-COPY build_scripts/manylinux-check.py /build_scripts/manylinux-check.py
-COPY build_scripts/requirements.txt /build_scripts/requirements.txt
-COPY build_scripts/requirements-tools.txt /build_scripts/requirements-tools.txt
+COPY build_scripts/finalize.sh build_scripts/python-tag-abi-tag.py build_scripts/requirements.txt build_scripts/requirements-tools.txt /build_scripts/
 RUN manylinux-entrypoint /build_scripts/finalize.sh && rm -rf /build_scripts
 
 ENV SSL_CERT_FILE=/opt/_internal/certs.pem
diff --git a/docker/build_scripts/finalize.sh b/docker/build_scripts/finalize.sh
index a6556dd6..d436c2ec 100755
--- a/docker/build_scripts/finalize.sh
+++ b/docker/build_scripts/finalize.sh
@@ -52,15 +52,6 @@ export SSL_CERT_FILE=/opt/_internal/certs.pem
 # Deactivate the tools virtual environment
 deactivate
 
-
-for PYTHON in /opt/python/*/bin/python; do
-	# Smoke test to make sure that our Pythons work, and do indeed detect as
-	# being manylinux compatible:
-	$PYTHON $MY_DIR/manylinux-check.py ${AUDITWHEEL_POLICY} ${AUDITWHEEL_ARCH}
-	# Make sure that SSL cert checking works
-	$PYTHON $MY_DIR/ssl-check.py
-done
-
 # We do not need the precompiled .pyc and .pyo files.
 clean_pyc /opt/_internal
 
@@ -68,4 +59,3 @@ clean_pyc /opt/_internal
 rm -rf /root/.cache
 
 hardlink -cv /opt/_internal
-
diff --git a/docker/build_scripts/manylinux-check.py b/tests/manylinux-check.py
similarity index 100%
rename from docker/build_scripts/manylinux-check.py
rename to tests/manylinux-check.py
diff --git a/tests/run_tests.sh b/tests/run_tests.sh
new file mode 100755
index 00000000..cad79905
--- /dev/null
+++ b/tests/run_tests.sh
@@ -0,0 +1,25 @@
+#!/bin/bash
+
+# Stop at any error, show all commands
+set -exuo pipefail
+
+# Get script directory
+MY_DIR=$(dirname "${BASH_SOURCE[0]}")
+
+
+for PYTHON in /opt/python/*/bin/python; do
+	# Smoke test to make sure that our Pythons work, and do indeed detect as
+	# being manylinux compatible:
+	$PYTHON $MY_DIR/manylinux-check.py ${AUDITWHEEL_POLICY} ${AUDITWHEEL_ARCH}
+	# Make sure that SSL cert checking works
+	$PYTHON $MY_DIR/ssl-check.py
+done
+
+# minimal tests for tools that should be present
+autoconf --version
+automake --version
+libtoolize --version
+patchelf --version
+git --version
+cmake --version
+swig -version
diff --git a/docker/build_scripts/ssl-check.py b/tests/ssl-check.py
similarity index 100%
rename from docker/build_scripts/ssl-check.py
rename to tests/ssl-check.py