diff --git a/doc/source/whatsnew/v1.1.0.rst b/doc/source/whatsnew/v1.1.0.rst
index 20415bba99476..88468050f9b2a 100644
--- a/doc/source/whatsnew/v1.1.0.rst
+++ b/doc/source/whatsnew/v1.1.0.rst
@@ -85,6 +85,7 @@ Other API changes
 - Added :meth:`DataFrame.value_counts` (:issue:`5377`)
 - :meth:`Groupby.groups` now returns an abbreviated representation when called on large dataframes (:issue:`1135`)
 - ``loc`` lookups with an object-dtype :class:`Index` and an integer key will now raise ``KeyError`` instead of ``TypeError`` when key is missing (:issue:`31905`)
+- Using a :func:`pandas.api.indexers.BaseIndexer` with ``min``, ``max``, ``std``, ``var``, ``count``, ``skew``, ``cov``, ``corr`` will now raise a ``NotImplementedError`` (:issue:`32865`)
 -
 
 Backwards incompatible API changes
diff --git a/pandas/core/window/common.py b/pandas/core/window/common.py
index fcde494f7f751..8abc47886d261 100644
--- a/pandas/core/window/common.py
+++ b/pandas/core/window/common.py
@@ -323,3 +323,13 @@ def func(arg, window, min_periods=None):
         return cfunc(arg, window, min_periods)
 
     return func
+
+
+def validate_baseindexer_support(func_name: Optional[str]) -> None:
+    # GH 32865: These functions work correctly with a BaseIndexer subclass
+    BASEINDEXER_WHITELIST = {"mean", "sum", "median", "kurt", "quantile"}
+    if isinstance(func_name, str) and func_name not in BASEINDEXER_WHITELIST:
+        raise NotImplementedError(
+            f"{func_name} is not supported with using a BaseIndexer "
+            f"subclasses. You can use .apply() with {func_name}."
+        )
diff --git a/pandas/core/window/rolling.py b/pandas/core/window/rolling.py
index aeab51149ec4e..dc8cf839d0bcb 100644
--- a/pandas/core/window/rolling.py
+++ b/pandas/core/window/rolling.py
@@ -46,6 +46,7 @@
     calculate_center_offset,
     calculate_min_periods,
     get_weighted_roll_func,
+    validate_baseindexer_support,
     zsqrt,
 )
 from pandas.core.window.indexers import (
@@ -391,11 +392,12 @@ def _get_cython_func_type(self, func: str) -> Callable:
             return self._get_roll_func(f"{func}_variable")
         return partial(self._get_roll_func(f"{func}_fixed"), win=self._get_window())
 
-    def _get_window_indexer(self, window: int) -> BaseIndexer:
+    def _get_window_indexer(self, window: int, func_name: Optional[str]) -> BaseIndexer:
         """
         Return an indexer class that will compute the window start and end bounds
         """
         if isinstance(self.window, BaseIndexer):
+            validate_baseindexer_support(func_name)
             return self.window
         if self.is_freq_type:
             return VariableWindowIndexer(index_array=self._on.asi8, window_size=window)
@@ -441,7 +443,7 @@ def _apply(
 
         blocks, obj = self._create_blocks()
         block_list = list(blocks)
-        window_indexer = self._get_window_indexer(window)
+        window_indexer = self._get_window_indexer(window, name)
 
         results = []
         exclude: List[Scalar] = []
@@ -1173,6 +1175,8 @@ class _Rolling_and_Expanding(_Rolling):
     )
 
     def count(self):
+        if isinstance(self.window, BaseIndexer):
+            validate_baseindexer_support("count")
 
         blocks, obj = self._create_blocks()
         results = []
@@ -1627,6 +1631,9 @@ def quantile(self, quantile, interpolation="linear", **kwargs):
     """
 
     def cov(self, other=None, pairwise=None, ddof=1, **kwargs):
+        if isinstance(self.window, BaseIndexer):
+            validate_baseindexer_support("cov")
+
         if other is None:
             other = self._selected_obj
             # only default unset
@@ -1770,6 +1777,9 @@ def _get_cov(X, Y):
     )
 
     def corr(self, other=None, pairwise=None, **kwargs):
+        if isinstance(self.window, BaseIndexer):
+            validate_baseindexer_support("corr")
+
         if other is None:
             other = self._selected_obj
             # only default unset
diff --git a/pandas/tests/window/test_base_indexer.py b/pandas/tests/window/test_base_indexer.py
index 606520c6d68ca..e9190dfde4fc4 100644
--- a/pandas/tests/window/test_base_indexer.py
+++ b/pandas/tests/window/test_base_indexer.py
@@ -80,3 +80,18 @@ def get_window_bounds(self, num_values, min_periods, center, closed):
     indexer = CustomIndexer()
     with pytest.raises(NotImplementedError, match="BaseIndexer subclasses not"):
         df.rolling(indexer, win_type="boxcar")
+
+
+@pytest.mark.parametrize(
+    "func", ["min", "max", "std", "var", "count", "skew", "cov", "corr"]
+)
+def test_notimplemented_functions(func):
+    # GH 32865
+    class CustomIndexer(BaseIndexer):
+        def get_window_bounds(self, num_values, min_periods, center, closed):
+            return np.array([0, 1]), np.array([1, 2])
+
+    df = DataFrame({"values": range(2)})
+    indexer = CustomIndexer()
+    with pytest.raises(NotImplementedError, match=f"{func} is not supported"):
+        getattr(df.rolling(indexer), func)()