diff --git a/muted-tests.yml b/muted-tests.yml
index 7ab855ac5c293..0616f4f92034a 100644
--- a/muted-tests.yml
+++ b/muted-tests.yml
@@ -209,9 +209,6 @@ tests:
   issue: https://github.com/elastic/elasticsearch/issues/120668
 - class: org.elasticsearch.xpack.security.authc.ldap.ADLdapUserSearchSessionFactoryTests
   issue: https://github.com/elastic/elasticsearch/issues/119882
-- class: org.elasticsearch.xpack.esql.action.CrossClusterAsyncEnrichStopIT
-  method: testEnrichAfterStop
-  issue: https://github.com/elastic/elasticsearch/issues/120757
 - class: org.elasticsearch.xpack.test.rest.XPackRestIT
   method: test {p0=ml/3rd_party_deployment/Test start deployment fails while model download in progress}
   issue: https://github.com/elastic/elasticsearch/issues/120810
diff --git a/x-pack/plugin/esql/compute/src/main/java/org/elasticsearch/compute/operator/exchange/ExchangeBuffer.java b/x-pack/plugin/esql/compute/src/main/java/org/elasticsearch/compute/operator/exchange/ExchangeBuffer.java
index ce400ddbdd6f9..23c98a1df193d 100644
--- a/x-pack/plugin/esql/compute/src/main/java/org/elasticsearch/compute/operator/exchange/ExchangeBuffer.java
+++ b/x-pack/plugin/esql/compute/src/main/java/org/elasticsearch/compute/operator/exchange/ExchangeBuffer.java
@@ -47,7 +47,17 @@ void addPage(Page page) {
             notifyNotEmpty();
         }
         if (noMoreInputs) {
-            discardPages();
+            // O(N) but acceptable because it only occurs with the stop API, and the queue size should be very small.
+            if (queue.removeIf(p -> p == page)) {
+                page.releaseBlocks();
+                final int size = queueSize.decrementAndGet();
+                if (size == maxSize - 1) {
+                    notifyNotFull();
+                }
+                if (size == 0) {
+                    completionFuture.onResponse(null);
+                }
+            }
         }
     }
 
diff --git a/x-pack/plugin/esql/compute/src/test/java/org/elasticsearch/compute/operator/exchange/ExchangeBufferTests.java b/x-pack/plugin/esql/compute/src/test/java/org/elasticsearch/compute/operator/exchange/ExchangeBufferTests.java
index bd5b53fb25c8b..7213e0b27aea0 100644
--- a/x-pack/plugin/esql/compute/src/test/java/org/elasticsearch/compute/operator/exchange/ExchangeBufferTests.java
+++ b/x-pack/plugin/esql/compute/src/test/java/org/elasticsearch/compute/operator/exchange/ExchangeBufferTests.java
@@ -66,6 +66,25 @@ public void testDrainPages() throws Exception {
         blockFactory.ensureAllBlocksAreReleased();
     }
 
+    public void testOutstandingPages() throws Exception {
+        ExchangeBuffer buffer = new ExchangeBuffer(randomIntBetween(1000, 10000));
+        var blockFactory = blockFactory();
+        Page p1 = randomPage(blockFactory);
+        Page p2 = randomPage(blockFactory);
+        buffer.addPage(p1);
+        buffer.addPage(p2);
+        buffer.finish(false);
+        buffer.addPage(randomPage(blockFactory));
+        assertThat(buffer.size(), equalTo(2));
+        assertSame(buffer.pollPage(), p1);
+        p1.releaseBlocks();
+        assertSame(buffer.pollPage(), p2);
+        p2.releaseBlocks();
+        assertNull(buffer.pollPage());
+        assertTrue(buffer.isFinished());
+        blockFactory.ensureAllBlocksAreReleased();
+    }
+
     private static MockBlockFactory blockFactory() {
         BigArrays bigArrays = new MockBigArrays(PageCacheRecycler.NON_RECYCLING_INSTANCE, ByteSizeValue.ofGb(1)).withCircuitBreaking();
         CircuitBreaker breaker = bigArrays.breakerService().getBreaker(CircuitBreaker.REQUEST);