diff --git a/django/db/migrations/executor.py b/django/db/migrations/executor.py
index 4aaa91ef33..7b779819e9 100644
--- a/django/db/migrations/executor.py
+++ b/django/db/migrations/executor.py
@@ -25,9 +25,9 @@ class MigrationExecutor:
         """
         plan = []
         if clean_start:
-            applied = set()
+            applied = {}
         else:
-            applied = set(self.loader.applied_migrations)
+            applied = dict(self.loader.applied_migrations)
         for target in targets:
             # If the target is (app_label, None), that means unmigrate everything
             if target[1] is None:
@@ -36,7 +36,7 @@ class MigrationExecutor:
                         for migration in self.loader.graph.backwards_plan(root):
                             if migration in applied:
                                 plan.append((self.loader.graph.nodes[migration], True))
-                                applied.remove(migration)
+                                applied.pop(migration)
             # If the migration is already applied, do backwards mode,
             # otherwise do forwards mode.
             elif target in applied:
@@ -53,12 +53,12 @@ class MigrationExecutor:
                     for migration in self.loader.graph.backwards_plan(node):
                         if migration in applied:
                             plan.append((self.loader.graph.nodes[migration], True))
-                            applied.remove(migration)
+                            applied.pop(migration)
             else:
                 for migration in self.loader.graph.forwards_plan(target):
                     if migration not in applied:
                         plan.append((self.loader.graph.nodes[migration], False))
-                        applied.add(migration)
+                        applied[migration] = self.loader.graph.nodes[migration]
         return plan
 
     def _create_project_state(self, with_applied_migrations=False):
diff --git a/django/db/migrations/loader.py b/django/db/migrations/loader.py
index 54e1cefb00..8c308621d2 100644
--- a/django/db/migrations/loader.py
+++ b/django/db/migrations/loader.py
@@ -206,7 +206,7 @@ class MigrationLoader:
         self.load_disk()
         # Load database data
         if self.connection is None:
-            self.applied_migrations = set()
+            self.applied_migrations = {}
         else:
             recorder = MigrationRecorder(self.connection)
             self.applied_migrations = recorder.applied_migrations()
@@ -232,9 +232,9 @@ class MigrationLoader:
             # Ensure the replacing migration is only marked as applied if all of
             # its replacement targets are.
             if all(applied_statuses):
-                self.applied_migrations.add(key)
+                self.applied_migrations[key] = migration
             else:
-                self.applied_migrations.discard(key)
+                self.applied_migrations.pop(key, None)
             # A replacing migration can be used if either all or none of its
             # replacement targets have been applied.
             if all(applied_statuses) or (not any(applied_statuses)):
diff --git a/django/db/migrations/recorder.py b/django/db/migrations/recorder.py
index ad5435d906..c3ed4148e7 100644
--- a/django/db/migrations/recorder.py
+++ b/django/db/migrations/recorder.py
@@ -69,13 +69,16 @@ class MigrationRecorder:
             raise MigrationSchemaMissing("Unable to create the django_migrations table (%s)" % exc)
 
     def applied_migrations(self):
-        """Return a set of (app, name) of applied migrations."""
+        """
+        Return a dict mapping (app_name, migration_name) to Migration instances
+        for all applied migrations.
+        """
         if self.has_table():
-            return {tuple(x) for x in self.migration_qs.values_list('app', 'name')}
+            return {(migration.app, migration.name): migration for migration in self.migration_qs}
         else:
             # If the django_migrations table doesn't exist, then no migrations
             # are applied.
-            return set()
+            return {}
 
     def record_applied(self, app, name):
         """Record that a migration was applied."""
diff --git a/tests/migrations/test_executor.py b/tests/migrations/test_executor.py
index 8bb4f83c5f..ddba8d12e8 100644
--- a/tests/migrations/test_executor.py
+++ b/tests/migrations/test_executor.py
@@ -711,7 +711,11 @@ class ExecutorUnitTests(SimpleTestCase):
         graph.add_dependency(None, a2, a1)
 
         executor = MigrationExecutor(None)
-        executor.loader = FakeLoader(graph, {a1, b1, a2})
+        executor.loader = FakeLoader(graph, {
+            a1: a1_impl,
+            b1: b1_impl,
+            a2: a2_impl,
+        })
 
         plan = executor.migration_plan({a1})
 
@@ -754,7 +758,14 @@ class ExecutorUnitTests(SimpleTestCase):
         graph.add_dependency(None, b2, a2)
 
         executor = MigrationExecutor(None)
-        executor.loader = FakeLoader(graph, {a1, b1, a2, b2, a3, a4})
+        executor.loader = FakeLoader(graph, {
+            a1: a1_impl,
+            b1: b1_impl,
+            a2: a2_impl,
+            b2: b2_impl,
+            a3: a3_impl,
+            a4: a4_impl,
+        })
 
         plan = executor.migration_plan({a1})
 
@@ -791,7 +802,10 @@ class ExecutorUnitTests(SimpleTestCase):
         graph.add_dependency(None, c1, a1)
 
         executor = MigrationExecutor(None)
-        executor.loader = FakeLoader(graph, {a1, b1})
+        executor.loader = FakeLoader(graph, {
+            a1: a1_impl,
+            b1: b1_impl,
+        })
 
         plan = executor.migration_plan({a1})
 
diff --git a/tests/migrations/test_loader.py b/tests/migrations/test_loader.py
index e3a635dc63..0f0a590e7c 100644
--- a/tests/migrations/test_loader.py
+++ b/tests/migrations/test_loader.py
@@ -265,7 +265,7 @@ class LoaderTests(TestCase):
 
         def num_nodes():
             plan = set(loader.graph.forwards_plan(('migrations', '7_auto')))
-            return len(plan - loader.applied_migrations)
+            return len(plan - loader.applied_migrations.keys())
 
         # Empty database: use squashed migration
         loader.build_graph()
@@ -339,7 +339,7 @@ class LoaderTests(TestCase):
         loader.build_graph()
 
         plan = set(loader.graph.forwards_plan(('app1', '4_auto')))
-        plan = plan - loader.applied_migrations
+        plan = plan - loader.applied_migrations.keys()
         expected_plan = {
             ('app2', '1_squashed_2'),
             ('app1', '3_auto'),
@@ -358,7 +358,7 @@ class LoaderTests(TestCase):
 
         def num_nodes():
             plan = set(loader.graph.forwards_plan(('migrations', '7_auto')))
-            return len(plan - loader.applied_migrations)
+            return len(plan - loader.applied_migrations.keys())
 
         # Empty database: use squashed migration
         loader.build_graph()
@@ -466,7 +466,7 @@ class LoaderTests(TestCase):
         # Load with nothing applied: both migrations squashed.
         loader.build_graph()
         plan = set(loader.graph.forwards_plan(('app1', '4_auto')))
-        plan = plan - loader.applied_migrations
+        plan = plan - loader.applied_migrations.keys()
         expected_plan = {
             ('app1', '1_auto'),
             ('app2', '1_squashed_2'),
@@ -480,7 +480,7 @@ class LoaderTests(TestCase):
         recorder.record_applied('app1', '2_auto')
         loader.build_graph()
         plan = set(loader.graph.forwards_plan(('app1', '4_auto')))
-        plan = plan - loader.applied_migrations
+        plan = plan - loader.applied_migrations.keys()
         expected_plan = {
             ('app2', '1_squashed_2'),
             ('app1', '3_auto'),
@@ -492,7 +492,7 @@ class LoaderTests(TestCase):
         recorder.record_applied('app2', '1_auto')
         loader.build_graph()
         plan = set(loader.graph.forwards_plan(('app1', '4_auto')))
-        plan = plan - loader.applied_migrations
+        plan = plan - loader.applied_migrations.keys()
         expected_plan = {
             ('app2', '2_auto'),
             ('app1', '3_auto'),