diff --git a/integration/config/config_test.go b/integration/config/config_test.go
index 275ba0dcc3..6ebfd2df8b 100644
--- a/integration/config/config_test.go
+++ b/integration/config/config_test.go
@@ -333,15 +333,20 @@ func TestTemplatedConfig(t *testing.T) {
 
 	outBuf.Reset()
 	errBuf.Reset()
+	// Read the mount table rather than running mount(8): busybox's mount stops
+	// parsing at an over-long entry and drops it along with everything after it,
+	// and the task carries one - the init binary is bind-mounted from an overlay
+	// whose options list every lower directory. /proc/mounts is what mount(8)
+	// reads anyway, and cat doesn't parse it.
 	resp = swarm.ExecTask(ctx, t, d, tasks[0], client.ExecCreateOptions{
-		Cmd:          []string{"mount"},
+		Cmd:          []string{"cat", "/proc/mounts"},
 		AttachStdout: true,
 		AttachStderr: true,
 	})
 
 	_, err = stdcopy.StdCopy(&outBuf, &errBuf, resp.Reader)
 	assert.NilError(t, err)
-	assert.Check(t, is.Contains(outBuf.String(), "tmpfs on /templated_config type tmpfs"), "expected to be mounted as tmpfs")
+	assert.Check(t, is.Contains(outBuf.String(), "tmpfs /templated_config tmpfs"), "expected to be mounted as tmpfs")
 	assert.Check(t, is.Equal(errBuf.String(), ""))
 }
 
diff --git a/integration/internal/swarm/service.go b/integration/internal/swarm/service.go
index 5035264737..603fb5e04d 100644
--- a/integration/internal/swarm/service.go
+++ b/integration/internal/swarm/service.go
@@ -68,13 +68,23 @@ func CreateService(ctx context.Context, t *testing.T, d *daemon.Daemon, opts ...
 	return resp.ID
 }
 
-// CreateServiceSpec creates a default service-spec, and applies the provided options
+// CreateServiceSpec creates a default service-spec, and applies the provided options.
+//
+// The defaults include running tasks under an init, so a test that wants the
+// daemon's own default instead has to say so with ServiceWithInit(nil) - saying
+// nothing gets the init.
 func CreateServiceSpec(t *testing.T, opts ...ServiceSpecOpt) swarmtypes.ServiceSpec {
 	t.Helper()
 	var spec swarmtypes.ServiceSpec
 	ServiceWithImage("busybox:latest")(&spec)
 	ServiceWithCommand([]string{"/bin/top"})(&spec)
 	ServiceWithReplicas(1)(&spec)
+	// Run tasks under an init. None of the commands these tests run install a
+	// SIGTERM handler, and a signal sent to PID 1 has no default action - so as
+	// PID 1 they ignore the stop signal, and every task teardown waits out the
+	// container's whole stop timeout before the kill. An init takes PID 1 and
+	// forwards the signal to a process the default action applies to.
+	ServiceWithInit(new(true))(&spec)
 
 	for _, o := range opts {
 		o(&spec)
diff --git a/integration/network/bridge/iptablesdoc/iptablesdoc_linux_test.go b/integration/network/bridge/iptablesdoc/iptablesdoc_linux_test.go
index 53eaa48b7f..6849667f34 100644
--- a/integration/network/bridge/iptablesdoc/iptablesdoc_linux_test.go
+++ b/integration/network/bridge/iptablesdoc/iptablesdoc_linux_test.go
@@ -366,6 +366,8 @@ func createServices(ctx context.Context, t *testing.T, d *daemon.Daemon, section
 						ContainerSpec: &swarmtypes.ContainerSpec{
 							Image:   "busybox:latest",
 							Command: []string{"/bin/top"},
+							// top ignores SIGTERM as PID 1; an init forwards it.
+							Init: new(true),
 						},
 					},
 					EndpointSpec: &swarmtypes.EndpointSpec{
diff --git a/integration/secret/secret_test.go b/integration/secret/secret_test.go
index 1c2dc7a4da..91a559fa9f 100644
--- a/integration/secret/secret_test.go
+++ b/integration/secret/secret_test.go
@@ -355,15 +355,20 @@ func TestTemplatedSecret(t *testing.T) {
 
 	outBuf.Reset()
 	errBuf.Reset()
+	// Read the mount table rather than running mount(8): busybox's mount stops
+	// parsing at an over-long entry and drops it along with everything after it,
+	// and the task carries one - the init binary is bind-mounted from an overlay
+	// whose options list every lower directory. /proc/mounts is what mount(8)
+	// reads anyway, and cat doesn't parse it.
 	resp = swarm.ExecTask(ctx, t, d, tasks[0], client.ExecCreateOptions{
-		Cmd:          []string{"mount"},
+		Cmd:          []string{"cat", "/proc/mounts"},
 		AttachStdout: true,
 		AttachStderr: true,
 	})
 
 	_, err = stdcopy.StdCopy(&outBuf, &errBuf, resp.Reader)
 	assert.NilError(t, err)
-	assert.Check(t, is.Contains(outBuf.String(), "tmpfs on /run/secrets/templated_secret type tmpfs"), "expected to be mounted as tmpfs")
+	assert.Check(t, is.Contains(outBuf.String(), "tmpfs /run/secrets/templated_secret tmpfs"), "expected to be mounted as tmpfs")
 	assert.Check(t, is.Equal(errBuf.String(), ""))
 }
 
diff --git a/integration/service/create_test.go b/integration/service/create_test.go
index 8ca6d6b8b1..f7c4018e81 100644
--- a/integration/service/create_test.go
+++ b/integration/service/create_test.go
@@ -44,7 +44,10 @@ func testServiceCreateInit(ctx context.Context, daemonEnabled bool) func(t *test
 		booleanTrue := true
 		booleanFalse := false
 
-		serviceID := swarm.CreateService(ctx, t, d)
+		// An unset Init delegates to the daemon configuration, which is what this
+		// case is about - so ask for it explicitly, rather than relying on the
+		// spec helper leaving it unset.
+		serviceID := swarm.CreateService(ctx, t, d, swarm.ServiceWithInit(nil))
 		poll.WaitOn(t, swarm.RunningTasksCount(ctx, apiClient, serviceID, 1), swarm.ServicePoll)
 		i := inspectServiceContainer(ctx, t, apiClient, serviceID)
 		// HostConfig.Init == nil means that it delegates to daemon configuration