O>>inputObservable
O>>      .Throttle(500)
O>>      .CombineLatest(input => GetList(input))
O>>      .OnNext( result => texbBox.text = result );
O>>

   Observable<string[]> inputObservable = new Observable<string[]>();

   // любое место где инициализация формы
   void Init()
   {
    inputObservable
         .Throttle(500) 
         .CombineLatest(input => GetList(input))
         .Subscribe( result => texbBox.text = result );
   }

   void KeyDown(EventArgs e) => inputObservable.OnNext(e);

 private volatile int _lastRequestId;

        private void textBox1_KeyUp(object sender, KeyEventArgs e)
        {
            _lastRequestId++;
            FillItems();
        }

        private async void FillItems()
        {
            var requestId = _lastRequestId;

            await Task.Delay(500);

            if (requestId != _lastRequestId)
                return;

            textBox2.Clear();

            var items = await MakeRequest(textBox1.Text);

            if (_lastRequestId != requestId)
                return;

            textBox2.Text = string.Join("\r\n", items);
        }

textBox2.Clear();

S>textBox2.Clear();
S>

...
.Throttle(500)
.Subscribe( input => textBox2.Clear() )
.CombineLatest( input => GetList(input))
.Subscribe( result => textBox2.text = result );

Observable<string[]> inputObservable = new Observable<string[]>();

using System;
using System.Collections.Generic;
using System.Reactive.Linq;
using System.Threading;
using System.Threading.Tasks;
using System.Windows.Forms;

namespace RxForumSample
{
    public partial class Form1 : Form
    {
        public Form1()
        {
            InitializeComponent();
        }

        private void Form1_Load(object sender, EventArgs e)
        {
            var observable = Observable.FromEventPattern(textBox1, "KeyUp");

            var cancelTokenSource = new CancellationTokenSource();

            var subscription = observable
                .Select(_ =>
                {
                    cancelTokenSource.Cancel();
                    cancelTokenSource = new CancellationTokenSource();
                    return cancelTokenSource;
                })
                .Throttle(TimeSpan.FromMilliseconds(500))
                .Select(async _ =>
                {
                    try
                    {
                        return await MakeRequest(textBox1.Text, _.Token);
                    }
                    catch (Exception exception)
                    {
                        Console.WriteLine(exception);
                        return null;
                    }
                })
                .Concat()
                .ObserveOn(textBox2)
                .Subscribe(items =>
                {
                    if (items != null)
                        textBox2.Text = string.Join("\r\n", items);
                });

        }

        
        Random _random = new Random();

        private async Task<List<string>> MakeRequest(string q, CancellationToken cancellationToken)
        {
            await Task.Delay(_random.Next(100, 5000), cancellationToken);

            Console.Beep(1000, 10);

            return await Task.FromResult(new List<string>()
            {
                q + " пункт 1 ",
                q + " пункт 2",
                q + " пункт 3",
            });

        }
    }
}

.ObserveOn(textBox2)