Stateful Children are updated by argument as well

main.dart

import 'package:flutter/material.dart';

final Color darkBlue = Color.fromARGB(255, 18, 32, 47);

void main() {
  runApp(MyApp());
}

class MyApp extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    return MaterialApp(
      theme: ThemeData.dark().copyWith(scaffoldBackgroundColor: darkBlue),
      debugShowCheckedModeBanner: false,
      home: Scaffold(
        body: Center(
          child: ParentWidget(),
        ),
      ),
    );
  }
}

class ParentWidget extends StatefulWidget {
  @override
  _ParentWidgetState createState() => _ParentWidgetState();
}

class _ParentWidgetState extends State<ParentWidget> {
  String arg = 'arg';

  @override
  Widget build(BuildContext context) {
    return Column(
      children: [
        ElevatedButton(
          onPressed: () {
            setState(() {
              arg += arg;
            });
          },
          child: Text('add arg'),
        ),
        OtherWidget(arg)
      ],
    );
  }
}

class OtherWidget extends StatefulWidget {
  OtherWidget(this.arg);

  final String arg;

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

class _OtherWidgetState extends State<OtherWidget> {
  @override
  void initState() {
    print('started state');
    super.initState();
  }

  @override
  void didChangeDependencies() {
    print('changed deps');
    super.didChangeDependencies();
  }

  @override
  void didUpdateWidget(covariant OtherWidget oldWidget) {
    print('updt widget');
    super.didUpdateWidget(oldWidget);
  }
  
  @override
  Widget build(BuildContext context) {
    return Container(
      child: Text(widget.arg),
    );
  }
}