diff --git a/kubert/src/shutdown.rs b/kubert/src/shutdown.rs
index 927ff76..9842b13 100644
--- a/kubert/src/shutdown.rs
+++ b/kubert/src/shutdown.rs
@@ -119,6 +119,9 @@ impl Shutdown {
 #[cfg(feature = "runtime")]
 impl<T> CancelOnShutdown<T> {
     /// Wraps a `Future` or `Stream` that completes when the shutdown watch fires.
+    ///
+    /// The inner `Future`/`Stream` is given the chance to complete before the shutdown watch is
+    /// polled so that it has a chance to complete its work before the task is cancelled.
     pub(crate) fn new(watch: Watch, inner: T) -> Self {
         // XXX Unfortunately the `Watch` API doesn't give us any means to poll for updates, so we
         // have to box the async call to poll it from the stream.
@@ -136,11 +139,14 @@ impl<F: std::future::Future<Output = ()>> std::future::Future for CancelOnShutdo
     fn poll(self: Pin<&mut Self>, cx: &mut Context<'_>) -> Poll<()> {
         let mut this = self.project();
 
-        if this.shutdown.as_mut().poll(cx).is_ready() {
+        // Drive the future to completion.
+        if this.inner.poll(cx).is_ready() {
             return Poll::Ready(());
         }
 
-        this.inner.poll(cx)
+        // If the future is pending, register interest in the shutdown watch and complete the future
+        // if it has fired.
+        this.shutdown.as_mut().poll(cx)
     }
 }
 
@@ -153,10 +159,66 @@ impl<S: futures_core::Stream> futures_core::Stream for CancelOnShutdown<S> {
 
         let mut this = self.project();
 
+        // Process items from the stream until it is pending (or the stream ends).
+        if let Poll::Ready(next) = this.inner.poll_next(cx) {
+            return Poll::Ready(next);
+        }
+
+        // If the stream is pending, register interest in the shutdown watch and end the stream if
+        // it has fired.
         if this.shutdown.as_mut().poll(cx).is_ready() {
             return Poll::Ready(None);
         }
 
-        this.inner.poll_next(cx)
+        Poll::Pending
+    }
+}
+
+#[cfg(all(test, feature = "runtime"))]
+mod test {
+    use super::CancelOnShutdown;
+    use tokio_stream::wrappers::ReceiverStream;
+    use tokio_test::{assert_pending, assert_ready, assert_ready_eq, task};
+
+    #[tokio::test]
+    async fn cancel_stream_drains() {
+        let (shutdown_tx, shutdown_rx) = drain::channel();
+
+        let (stream_tx, stream_rx) = tokio::sync::mpsc::channel(3);
+        let mut stream_rx = task::spawn(CancelOnShutdown::new(
+            shutdown_rx,
+            ReceiverStream::new(stream_rx),
+        ));
+        stream_tx.try_send(1).unwrap();
+        stream_tx.try_send(2).unwrap();
+        stream_tx.try_send(3).unwrap();
+
+        assert_ready_eq!(stream_rx.poll_next(), Some(1));
+
+        let mut drain = task::spawn(shutdown_tx.drain());
+        assert_ready_eq!(stream_rx.poll_next(), Some(2));
+        assert_ready_eq!(stream_rx.poll_next(), Some(3));
+        assert_pending!(drain.poll());
+        assert_ready_eq!(stream_rx.poll_next(), None);
+        assert_ready!(drain.poll());
+    }
+
+    #[tokio::test]
+    async fn cancel_future_ends() {
+        let (shutdown_tx, shutdown_rx) = drain::channel();
+
+        let (_tx, rx) = tokio::sync::oneshot::channel::<()>();
+        let mut rx = task::spawn(CancelOnShutdown::new(
+            shutdown_rx,
+            Box::pin(async move {
+                rx.await.unwrap();
+            }),
+        ));
+        assert_pending!(rx.poll());
+
+        let mut drain = task::spawn(shutdown_tx.drain());
+        assert_pending!(drain.poll());
+        assert_ready!(rx.poll());
+        assert_ready!(drain.poll());
     }
 }