fix: propagate Mount objects when using include_router

When an APIRouter with mounted sub-applications is included into a
FastAPI app via include_router(), the Mount routes were silently
ignored. This is because include_router() only handled APIRoute,
Route, APIWebSocketRoute, and WebSocketRoute — never Mount.

Additionally, APIRouter.mount() (inherited from Starlette's Router)
does not apply self.prefix to the stored path, unlike add_api_route().
So the fix applies both the include_router prefix and the router's own
prefix when propagating Mount routes.

Fixes #10180

fastapi/routing.py

@@ -1489,6 +1489,8 @@ class APIRouter(routing.Router):
                 self.add_websocket_route(
                     prefix + route.path, route.endpoint, name=route.name
                 )
+            elif isinstance(route, routing.Mount):
+                self.mount(prefix + router.prefix + route.path, route.app, name=route.name)
         for handler in router.on_startup:
             self.add_event_handler("startup", handler)
         for handler in router.on_shutdown:

tests/test_router_mount.py

@@ -0,0 +1,106 @@
+"""Tests for mounting sub-applications under APIRouter (issue #10180)."""
+from fastapi import APIRouter, FastAPI
+from starlette.staticfiles import StaticFiles
+from starlette.testclient import TestClient
+
+
+def test_mount_subapp_under_router_with_prefix() -> None:
+    """APIRouter with prefix should propagate Mount to the main app."""
+    app = FastAPI()
+    api_router = APIRouter(prefix="/api")
+
+    @api_router.get("/app")
+    def read_main() -> dict:
+        return {"message": "Hello World from main app"}
+
+    subapi = FastAPI()
+
+    @subapi.get("/sub")
+    def read_sub() -> dict:
+        return {"message": "Hello World from sub API"}
+
+    api_router.mount("/subapi", subapi)
+    app.include_router(api_router)
+
+    client = TestClient(app)
+
+    # Regular route should still work
+    r = client.get("/api/app")
+    assert r.status_code == 200
+    assert r.json() == {"message": "Hello World from main app"}
+
+    # Mounted sub-app should be accessible under router prefix
+    r = client.get("/api/subapi/sub")
+    assert r.status_code == 200
+    assert r.json() == {"message": "Hello World from sub API"}
+
+
+def test_mount_subapp_under_router_without_prefix() -> None:
+    """APIRouter without prefix should propagate Mount correctly."""
+    app = FastAPI()
+    router = APIRouter()
+
+    subapi = FastAPI()
+
+    @subapi.get("/hello")
+    def hello() -> dict:
+        return {"msg": "hello"}
+
+    router.mount("/sub", subapi)
+    app.include_router(router)
+
+    client = TestClient(app)
+
+    r = client.get("/sub/hello")
+    assert r.status_code == 200
+    assert r.json() == {"msg": "hello"}
+
+
+def test_mount_subapp_with_include_router_prefix() -> None:
+    """include_router prefix should be combined with router prefix and mount path."""
+    app = FastAPI()
+    router = APIRouter(prefix="/api")
+
+    subapi = FastAPI()
+
+    @subapi.get("/test")
+    def test_endpoint() -> dict:
+        return {"msg": "test"}
+
+    router.mount("/sub", subapi)
+    app.include_router(router, prefix="/v1")
+
+    client = TestClient(app)
+
+    r = client.get("/v1/api/sub/test")
+    assert r.status_code == 200
+    assert r.json() == {"msg": "test"}
+
+
+def test_mount_preserves_regular_routes() -> None:
+    """Adding Mount support should not break existing APIRoute handling."""
+    app = FastAPI()
+    api_router = APIRouter(prefix="/api")
+
+    @api_router.get("/items")
+    def list_items() -> dict:
+        return {"items": [1, 2, 3]}
+
+    @api_router.post("/items")
+    def create_item() -> dict:
+        return {"created": True}
+
+    subapi = FastAPI()
+
+    @subapi.get("/status")
+    def status() -> dict:
+        return {"status": "ok"}
+
+    api_router.mount("/health", subapi)
+    app.include_router(api_router)
+
+    client = TestClient(app)
+
+    assert client.get("/api/items").status_code == 200
+    assert client.post("/api/items").status_code == 200
+    assert client.get("/api/health/status").status_code == 200