diff --git a/pandas/conftest.py b/pandas/conftest.py
index 2862f7c957abc..bf5e632374b59 100644
--- a/pandas/conftest.py
+++ b/pandas/conftest.py
@@ -473,7 +473,7 @@ def index_with_missing(request):
     Fixture for indices with missing values
     """
     if request.param in ["int", "uint", "range", "empty", "repeats"]:
-        pytest.xfail("missing values not supported")
+        pytest.skip("missing values not supported")
     # GH 35538. Use deep copy to avoid illusive bug on np-dev
     # Azure pipeline that writes into indices_dict despite copy
     ind = indices_dict[request.param].copy(deep=True)
diff --git a/pandas/tests/indexing/test_coercion.py b/pandas/tests/indexing/test_coercion.py
index 8735e2a09920d..ac2e300f9f8d6 100644
--- a/pandas/tests/indexing/test_coercion.py
+++ b/pandas/tests/indexing/test_coercion.py
@@ -436,7 +436,20 @@ def test_insert_index_float64(self, insert, coerced_val, coerced_dtype):
         ],
         ids=["datetime64", "datetime64tz"],
     )
-    def test_insert_index_datetimes(self, fill_val, exp_dtype):
+    @pytest.mark.parametrize(
+        "insert_value",
+        [pd.Timestamp("2012-01-01"), pd.Timestamp("2012-01-01", tz="Asia/Tokyo"), 1],
+    )
+    def test_insert_index_datetimes(self, request, fill_val, exp_dtype, insert_value):
+        if not hasattr(insert_value, "tz"):
+            request.node.add_marker(
+                pytest.mark.xfail(reason="ToDo: must coerce to object")
+            )
+        elif fill_val.tz != insert_value.tz:
+            request.node.add_marker(
+                pytest.mark.xfail(reason="GH 37605 - require tz equality?")
+            )
+
         obj = pd.DatetimeIndex(
             ["2011-01-01", "2011-01-02", "2011-01-03", "2011-01-04"], tz=fill_val.tz
         )
@@ -448,25 +461,7 @@ def test_insert_index_datetimes(self, fill_val, exp_dtype):
         )
         self._assert_insert_conversion(obj, fill_val, exp, exp_dtype)
 
-        if fill_val.tz:
-            msg = "Cannot compare tz-naive and tz-aware"
-            with pytest.raises(TypeError, match=msg):
-                obj.insert(1, pd.Timestamp("2012-01-01"))
-
-            msg = "Timezones don't match"
-            with pytest.raises(ValueError, match=msg):
-                obj.insert(1, pd.Timestamp("2012-01-01", tz="Asia/Tokyo"))
-
-        else:
-            msg = "Cannot compare tz-naive and tz-aware"
-            with pytest.raises(TypeError, match=msg):
-                obj.insert(1, pd.Timestamp("2012-01-01", tz="Asia/Tokyo"))
-
-        msg = "value should be a 'Timestamp' or 'NaT'. Got 'int' instead."
-        with pytest.raises(TypeError, match=msg):
-            obj.insert(1, 1)
-
-        pytest.xfail("ToDo: must coerce to object")
+        obj.insert(1, insert_value)
 
     def test_insert_index_timedelta64(self):
         obj = pd.TimedeltaIndex(["1 day", "2 day", "3 day", "4 day"])
diff --git a/pandas/tests/tseries/offsets/common.py b/pandas/tests/tseries/offsets/common.py
index b2ac28e1865d6..5edef896be537 100644
--- a/pandas/tests/tseries/offsets/common.py
+++ b/pandas/tests/tseries/offsets/common.py
@@ -98,12 +98,10 @@ def _get_offset(self, klass, value=1, normalize=False):
             klass = klass(value, normalize=normalize)
         return klass
 
-    def test_apply_out_of_range(self, tz_naive_fixture):
+    def test_apply_out_of_range(self, request, tz_naive_fixture):
         tz = tz_naive_fixture
         if self._offset is None:
             return
-        if isinstance(tz, tzlocal) and not IS64:
-            pytest.xfail(reason="OverflowError inside tzlocal past 2038")
 
         # try to create an out-of-bounds result timestamp; if we can't create
         # the offset skip
@@ -123,6 +121,13 @@ def test_apply_out_of_range(self, tz_naive_fixture):
             t = Timestamp("20080101", tz=tz)
             result = t + offset
             assert isinstance(result, datetime)
+
+            if isinstance(tz, tzlocal) and not IS64:
+                # If we hit OutOfBoundsDatetime on non-64 bit machines
+                # we'll drop out of the try clause before the next test
+                request.node.add_marker(
+                    pytest.mark.xfail(reason="OverflowError inside tzlocal past 2038")
+                )
             assert t.tzinfo == result.tzinfo
 
         except OutOfBoundsDatetime: