Navigating to a new screen when stream value in BLOC changes

You should not use StreamBuilder to handle navigation.
StreamBuilder is used to build the content of a screen and nothing else.

Instead, you will have to listen to the stream to trigger side-effects manually. This is done by using a StatefulWidget and overriding initState/dispose as such:

class Example extends StatefulWidget {
  final Stream<int> stream;

  const Example({Key key, this.stream}) : super(key: key);

  @override
  ExampleState createState() => ExampleState();
}

class ExampleState extends State<Example> {
  StreamSubscription _streamSubscription;

  @override
  void initState() {
    super.initState();
    _listen();
  }

  @override
  void didUpdateWidget(Example oldWidget) {
    super.didUpdateWidget(oldWidget);
    if (oldWidget.stream != widget.stream) {
      _streamSubscription.cancel();
      _listen();
    }
  }

  void _listen() {
    _streamSubscription = widget.stream.listen((value) {
      Navigator.pushNamed(context, '/someRoute/$value');
    });
  }

  @override
  void dispose() {
    _streamSubscription.cancel();
    super.dispose();
  }

  @override
  Widget build(BuildContext context) {
    return Container();
  }
}

---

Note that if you’re using an InheritedWidget to obtain your stream (typically BLoC), you will want to use didChangeDependencies instead of initState/didUpdateWidget.

This leads to:

class Example extends StatefulWidget {
  @override
  ExampleState createState() => ExampleState();
}

class ExampleState extends State<Example> {
  StreamSubscription _streamSubscription;
  Stream _previousStream;

  void _listen(Stream<int> stream) {
    _streamSubscription = stream.listen((value) {
      Navigator.pushNamed(context, '/someRoute/$value');
    });
  }

  @override
  void didChangeDependencies() {
    super.didChangeDependencies();
    final bloc = MyBloc.of(context);
    if (bloc.stream != _previousStream) {
      _streamSubscription?.cancel();
      _previousStream = bloc.stream;
      _listen(bloc.stream);
    }
  }

  @override
  void dispose() {
    _streamSubscription.cancel();
    super.dispose();
  }

  @override
  Widget build(BuildContext context) {
    return Container();
  }
}