This Flutter app example show how to use bottom navigation bar with tabs inner navigation. Also, how to handle Android's back button to pop inner screens of tab and double tap bottom navigation item to pop all inner screens of a tab.

flutter_tab_navigator.dart

///
/// Example GIF image:
/// https://ibb.co/mbqF72Q
///


import 'package:flutter/material.dart';
import 'package:flutter/services.dart';

void main() {
  SystemChrome.setSystemUIOverlayStyle(SystemUiOverlayStyle(
    statusBarColor: Colors.transparent,
    statusBarBrightness: Brightness.dark,
    statusBarIconBrightness: Brightness.dark,
    systemNavigationBarColor: Colors.transparent,
    systemNavigationBarIconBrightness: Brightness.dark,
  ));

  runApp(MyApp());
}

class MyApp extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    return MaterialApp(
      title: 'My Flutter App',
      theme: ThemeData(
        primarySwatch: Colors.red,
        canvasColor: Colors.white,
      ),
      home: DashboardScreen(),
    );
  }
}

class DashboardScreen extends StatefulWidget {
  @override
  _DashboardScreenState createState() => _DashboardScreenState();
}

class _DashboardScreenState extends State<DashboardScreen> {
  final _tabNavigator = GlobalKey<TabNavigatorState>();
  final _tab1 = GlobalKey<NavigatorState>();
  final _tab2 = GlobalKey<NavigatorState>();
  final _tab3 = GlobalKey<NavigatorState>();

  var _tabSelectedIndex = 0;
  var _tabPopStack = false;

  void _setIndex(index) {
    setState(() {
      _tabPopStack = _tabSelectedIndex == index;
      _tabSelectedIndex = index;
    });
  }

  @override
  Widget build(BuildContext context) {
    return WillPopScope(
      onWillPop: () async => !await _tabNavigator.currentState.maybePop(),
      child: Scaffold(
        body: TabNavigator(
          key: _tabNavigator,
          tabs: <TabItem>[
            TabItem(_tab1, PageWithButton(title: 'Audio')),
            TabItem(_tab2, PageWithButton(title: 'Video')),
            TabItem(_tab3, PageWithButton(title: 'More')),
          ],
          selectedIndex: _tabSelectedIndex,
          popStack: _tabPopStack,
        ),
        bottomNavigationBar: BottomNavigationBar(
          currentIndex: _tabSelectedIndex,
          onTap: _setIndex,
          items: [
            BottomNavigationBarItem(
              icon: Icon(Icons.audiotrack),
              title: Text('Audio'),
            ),
            BottomNavigationBarItem(
              icon: Icon(Icons.ondemand_video),
              title: Text('Video'),
            ),
            BottomNavigationBarItem(
              icon: Icon(Icons.more_horiz),
              title: Text('More'),
            ),
          ],
        ),
      ),
    );
  }
}

class PageWithButton extends StatelessWidget {
  final String title;
  final int count;

  const PageWithButton({
    Key key,
    @required this.title,
    this.count = 0,
  }) : super(key: key);

  @override
  Widget build(BuildContext context) {
    return Scaffold(
      appBar: AppBar(
        title: Text(title),
        centerTitle: true,
      ),
      body: Center(
        child: RaisedButton(
          child: Text("$title $count"),
          onPressed: () {
            Navigator.of(context).push(MaterialPageRoute(
              builder: (_) => PageWithButton(title: title, count: count + 1),
            ));
          },
        ),
      ),
    );
  }
}

class TabItem {
  final GlobalKey<NavigatorState> key;
  final Widget tab;

  const TabItem(this.key, this.tab);
}

class TabNavigator extends StatefulWidget {
  final List<TabItem> tabs;
  final int selectedIndex;
  final bool popStack;

  TabNavigator({
    Key key,
    @required this.tabs,
    @required this.selectedIndex,
    this.popStack = false,
  }) : super(key: key);

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

class TabNavigatorState extends State<TabNavigator> {
  ///
  /// Try to pop widget, return true if popped
  ///
  Future<bool> maybePop() {
    return widget.tabs[widget.selectedIndex].key.currentState.maybePop();
  }

  _popStackIfRequired(BuildContext context) async {
    if (widget.popStack) {
      widget.tabs[widget.selectedIndex].key.currentState
          .popUntil((route) => route.isFirst);
    }
  }

  @override
  Widget build(BuildContext context) {
    print('selectedIndex=${widget.selectedIndex}, popStack=${widget.popStack}');

    _popStackIfRequired(context);

    return Stack(
      children: List.generate(widget.tabs.length, _buildTab),
    );
  }

  Widget _buildTab(int index) {
    return Offstage(
      offstage: widget.selectedIndex != index,
      child: Opacity(
        opacity: widget.selectedIndex == index ? 1.0 : 0.0,
        child: Navigator(
          key: widget.tabs[index].key,
          onGenerateRoute: (settings) => MaterialPageRoute(
                settings: settings,
                builder: (_) => widget.tabs[index].tab,
              ),
        ),
      ),
    );
  }
}

I think you'll have to work with Scroll controller.