diff --git a/libcontainer/container_linux.go b/libcontainer/container_linux.go
index e10ca2332ff..ae5d4fb46b4 100644
--- a/libcontainer/container_linux.go
+++ b/libcontainer/container_linux.go
@@ -524,6 +524,7 @@ func (c *Container) newParentProcess(p *Process) (parentProcess, error) {
 				// runc-dmz.
 				exePath = "/proc/self/exe"
 				p.clonedExes = append(p.clonedExes, dmzExe)
+				logrus.Debug("runc-dmz: using runc-dmz") // used for tests
 			} else if errors.Is(err, dmz.ErrNoDmzBinary) {
 				logrus.Debug("runc-dmz binary not embedded in runc binary, falling back to /proc/self/exe clone")
 			} else if err != nil {
@@ -542,6 +543,7 @@ func (c *Container) newParentProcess(p *Process) (parentProcess, error) {
 			}
 			exePath = "/proc/self/fd/" + strconv.Itoa(int(safeExe.Fd()))
 			p.clonedExes = append(p.clonedExes, safeExe)
+			logrus.Debug("runc-dmz: using /proc/self/exe clone") // used for tests
 		}
 		// Just to make sure we don't run without protection.
 		if dmzExe == nil && safeExe == nil {
diff --git a/tests/integration/helpers.bash b/tests/integration/helpers.bash
index cd08fb2459f..7e6399a47b8 100755
--- a/tests/integration/helpers.bash
+++ b/tests/integration/helpers.bash
@@ -646,12 +646,16 @@ function teardown_bundle() {
 	remove_parent
 }
 
-function requires_kernel() {
+function is_kernel_gte() {
 	local major_required minor_required
 	major_required=$(echo "$1" | cut -d. -f1)
 	minor_required=$(echo "$1" | cut -d. -f2)
-	if [[ "$KERNEL_MAJOR" -lt $major_required || ("$KERNEL_MAJOR" -eq $major_required && "$KERNEL_MINOR" -lt $minor_required) ]]; then
-		skip "requires kernel $1"
+	[[ "$KERNEL_MAJOR" -gt $major_required || ("$KERNEL_MAJOR" -eq $major_required && "$KERNEL_MINOR" -ge $minor_required) ]]
+}
+
+function requires_kernel() {
+	if ! is_kernel_gte "$@"; then
+		skip "requires kernel >= $1"
 	fi
 }
 
diff --git a/tests/integration/run.bats b/tests/integration/run.bats
index 9f1f1d8bc74..baf91fb00cd 100644
--- a/tests/integration/run.bats
+++ b/tests/integration/run.bats
@@ -126,3 +126,37 @@ function teardown() {
 	[ "$status" -eq 0 ]
 	[ "$output" = "410" ]
 }
+
+@test "runc run [runc-dmz]" {
+	runc --debug run test_hello
+	[ "$status" -eq 0 ]
+	[[ "$output" = *"Hello World"* ]]
+	# We use runc-dmz if we can.
+	[[ "$output" = *"runc-dmz: using runc-dmz"* ]]
+}
+
+@test "runc run [cap_sys_ptrace -> /proc/self/exe clone]" {
+	# Add CAP_SYS_PTRACE to the bounding set, the minimum needed to indicate a
+	# container process _could_ get CAP_SYS_PTRACE.
+	update_config '.process.capabilities.bounding += ["CAP_SYS_PTRACE"]'
+
+	runc --debug run test_hello
+	[ "$status" -eq 0 ]
+	[[ "$output" = *"Hello World"* ]]
+	if [ "$EUID" -ne 0 ] && is_kernel_gte 4.10; then
+		# For Linux 4.10 and later, rootless containers will use runc-dmz
+		# because they are running in a user namespace. See isDmzBinarySafe().
+		[[ "$output" = *"runc-dmz: using runc-dmz"* ]]
+	else
+		# If the container has CAP_SYS_PTRACE and is not rootless, we use
+		# /proc/self/exe cloning.
+		[[ "$output" = *"runc-dmz: using /proc/self/exe clone"* ]]
+	fi
+}
+
+@test "RUNC_DMZ=legacy runc run [/proc/self/exe clone]" {
+	RUNC_DMZ=legacy runc --debug run test_hello
+	[ "$status" -eq 0 ]
+	[[ "$output" = *"Hello World"* ]]
+	[[ "$output" = *"runc-dmz: using /proc/self/exe clone"* ]]
+}
diff --git a/tests/integration/seccomp-notify-compat.bats b/tests/integration/seccomp-notify-compat.bats
index 8d663edda51..6ca3449bffa 100644
--- a/tests/integration/seccomp-notify-compat.bats
+++ b/tests/integration/seccomp-notify-compat.bats
@@ -3,8 +3,8 @@
 load helpers
 
 function setup() {
-	if [[ "$KERNEL_MAJOR" -gt 5 || ("$KERNEL_MAJOR" -eq 5 && "$KERNEL_MINOR" -ge 6) ]]; then
-		skip "requires kernel less than 5.6"
+	if is_kernel_gte 5.6; then
+		skip "requires kernel < 5.6"
 	fi
 
 	requires arch_x86_64