diff --git a/moto/cognitoidp/models.py b/moto/cognitoidp/models.py
index 6ee71cbc..7078583f 100644
--- a/moto/cognitoidp/models.py
+++ b/moto/cognitoidp/models.py
@@ -1066,5 +1066,7 @@ def find_region_by_value(key, value):
 
             if key == "access_token" and value in user_pool.access_tokens:
                 return region
-
-    return cognitoidp_backends.keys()[0]
+    # If we can't find the `client_id` or `access_token`, we just pass
+    # back a default backend region, which will raise the appropriate
+    # error message (e.g. NotAuthorized or NotFound).
+    return list(cognitoidp_backends)[0]
diff --git a/tests/test_cognitoidp/test_cognitoidp.py b/tests/test_cognitoidp/test_cognitoidp.py
index 54ee9528..c61be4aa 100644
--- a/tests/test_cognitoidp/test_cognitoidp.py
+++ b/tests/test_cognitoidp/test_cognitoidp.py
@@ -1840,6 +1840,31 @@ def test_admin_set_user_password():
     result["UserStatus"].should.equal("CONFIRMED")
 
 
+@mock_cognitoidp
+def test_change_password_with_invalid_token_raises_error():
+    client = boto3.client("cognito-idp", "us-west-2")
+    with pytest.raises(ClientError) as ex:
+        client.change_password(
+            AccessToken=str(uuid.uuid4()),
+            PreviousPassword="previous_password",
+            ProposedPassword="newer_password",
+        )
+    ex.value.response["Error"]["Code"].should.equal("NotAuthorizedException")
+
+
+@mock_cognitoidp
+def test_confirm_forgot_password_with_non_existent_client_id_raises_error():
+    client = boto3.client("cognito-idp", "us-west-2")
+    with pytest.raises(ClientError) as ex:
+        client.confirm_forgot_password(
+            ClientId="non-existent-client-id",
+            Username="not-existent-username",
+            ConfirmationCode=str(uuid.uuid4()),
+            Password=str(uuid.uuid4()),
+        )
+    ex.value.response["Error"]["Code"].should.equal("ResourceNotFoundException")
+
+
 # Test will retrieve public key from cognito.amazonaws.com/.well-known/jwks.json,
 # which isnt mocked in ServerMode
 if not settings.TEST_SERVER_MODE: