在ASP.NET Core中，WebSocket连接的ping和pong消息是由WebSocket协议自动处理的，不需要手动编写代码来处理。如果WebSocket连接的ping消息连续失败了一定次数，连接会自动关闭。

以下是一个简单的ASP.NET Core WebSocket示例，展示了如何使用WebSocket：

using Microsoft.AspNetCore.Builder;
using Microsoft.AspNetCore.Hosting;
using Microsoft.AspNetCore.Http;
using Microsoft.Extensions.Hosting;
using System;
using System.Net.WebSockets;
using System.Threading;
using System.Threading.Tasks;

public class Startup
{
    public void Configure(IApplicationBuilder app, IWebHostEnvironment env)
    {
        app.UseWebSockets();

        app.Use(async (context, next) =>
        {
            if (context.Request.Path == "/ws")
            {
                if (context.WebSockets.IsWebSocketRequest)
                {
                    WebSocket webSocket = await context.WebSockets.AcceptWebSocketAsync();
                    await HandleWebSocket(webSocket);
                }
                else
                {
                    context.Response.StatusCode = 400;
                }
            }
            else
            {
                await next();
            }
        });

        app.Run(async (context) =>
        {
            await context.Response.WriteAsync("Hello World!");
        });
    }

    private async Task HandleWebSocket(WebSocket webSocket)
    {
        byte[] buffer = new byte[1024];
        WebSocketReceiveResult result = await webSocket.ReceiveAsync(new ArraySegment(buffer), CancellationToken.None);

        while (!result.CloseStatus.HasValue)
        {
            await webSocket.SendAsync(new ArraySegment(buffer, 0, result.Count), result.MessageType, result.EndOfMessage, CancellationToken.None);
            result = await webSocket.ReceiveAsync(new ArraySegment(buffer), CancellationToken.None);
        }

        await webSocket.CloseAsync(result.CloseStatus.Value, result.CloseStatusDescription, CancellationToken.None);
    }
}

public class Program
{
    public static void Main(string[] args)
    {
        CreateHostBuilder(args).Build().Run();
    }

    public static IHostBuilder CreateHostBuilder(string[] args) =>
        Host.CreateDefaultBuilder(args)
            .ConfigureWebHostDefaults(webBuilder =>
            {
                webBuilder.UseStartup();
            });
}

在上述代码中，当收到WebSocket连接请求时，我们会通过调用AcceptWebSocketAsync方法来接受WebSocket连接，并将连接交给HandleWebSocket方法来处理。在HandleWebSocket方法中，我们使用ReceiveAsync方法来接收客户端发送的消息，并通过SendAsync方法将消息发送回客户端。如果接收到的消息是ping消息，则WebSocket协议会自动处理pong消息的发送。如果连续多次接收ping消息失败，连接会自动关闭。

注意：上述代码只是一个简单的示例，实际应用中可能需要根据具体需求进行修改和扩展。