How can I update ASP.NET Core app over an existing/running site without stopping the web server?

When running with IIS you can drop a file called app_offline.htm (case sensitive) to your application folder. IIS will stop your application and will serve the contents of the app_offline.html file. Now you can copy your application. Once finished remove the app_offline.html and IIS will start your app. This is described in the docs and also in my post on running Asp.NET Core apps with IIS.