In iOS, we have a UITabBarController which stays permanently at the bottom of the screen when we push to a new ViewController.
In Flutter, we have a bottomNavigation
You can create Navigator widget in a Stack widget to use BottomNavigationBar with tabs' inner navigation. You can use WillPopScope to handle Android's back button to pop inner screens of tab. Also, double tap bottom navigation item to pop all inner screens of a tab.
I've created a Sample app for this.
Hope this help!