Inherited web.configs with ASP.NET Core Equivalent - iis

Question

My company is exploring ASP.NET Core. We currently use ASP.NET Web API 2 & MVC 5.2.2. We deploy our applications under IIS using the following configuration:
VirtualDirectory1 (web.config with connection string and other non-specific information)
App1 (app specific web.config)
App2 (app specific web.config)
Api1 (api specific web.config)
Api2 (api specific web.config)
VirtualDirectory2 (web.config with connection string and other non-specific information)
App1 (app specific web.config)
App2 (app specific web.config)
Api1 (api specific web.config)
Api2 (api specific web.config)
As you know, each application or API inherits configuration from the parent web.config under IIS, so each inherits the parent connection string. We use a single copy of each application on the server. Can we deploy ASP.NET Core applications the same way?
I found StackOverflow articles on nested objects and environment specific AppSettings.json configurations but this does not appear to be the answer.
UPDATE 1:
The above is a multi-tenant configuration where each virtual directory is a tenant. The web.config at the root of each virtual directory contains the connection string all apps under the virtual directory use. 
I read that a web.config file must exist for ASP.NET Core apps to work under IIS.
https://learn.microsoft.com/en-us/aspnet/core/host-and-deploy/iis/index?tabs=aspnetcore2x#webconfig-file
https://learn.microsoft.com/en-us/aspnet/core/fundamentals/configuration/?tabs=basicconfiguration#webconfig-file
So I wonder if that the connection string is visible to the ASP.NET Core applications.
Update 2:
A.F.A.I.K, web.config can be loaded from the app content root, but the one we need resides up in a virtual directory. A AspNetCoreModule maintainer told me environment variable inheritance (which applies in this scenario) is unreliable, and no plumbing exists to access other inherited configuration. The web.config only exists to configure IIS for ASP.NET Core applications.
The best solution seems to be using host headers the way Saaskit does.
https://github.com/saaskit/saaskit
This requires us to change our IIS deployment and add a DNS record per tenant, but it is the next best thing to inherited web.config files. I would like the ASP.NET Core team to consider introducing a ASP.NET Core-approach to inherited settings that gives us the same ability inherited web.configs did.
Putting tenant specific settings in a file up in a virtual directory is out of the box thinking, but it greatly simplifies multi-tenant support. Our MVC & Web API code needs nothing extra. It simply retrieves settings through the ConfigurationManager as though no other tenant exists.
We have one website under IIS and one virtual directory per tenant. Each tenant is accessed via its URI.
https://myfqdn.com/tenant1/myapp
https://myfqdn.com/tenant2/myapp
tenant1 and tenant2 are virtual directories containing a tenant specific web.config, so invoking https://myfqdn.com/tenant1/myapp loads tenant1's web.config and myapp inherits its settings for requests. /tenant2/myapp loads a different web.config.
Myapp is unaware of which virtual directory it was invoked from on each request. We install myapp in one physical location but add it as an application under each tenant's virtual directory. The result is we have one copy of the app, multiple instances, and no multi-tenancy plumbing.

Answer 1

ASP.NET Core is not bound to IIS so the same approach won't work as for ASP.NET. But you can implement a similar approach if you would like by moving your common settings to some SharedSettings.json and use this.
.ConfigureAppConfiguration((hostingContext, config) =>
{
var env = hostingContext.HostingEnvironment;
var sharedFolder = Path.Combine(env.ContentRootPath, "..", "Shared");
config
.AddJsonFile(Path.Combine(sharedFolder, "SharedSettings.json"), optional: true) // When running using dotnet run
.AddJsonFile("SharedSettings.json", optional: true) // When app is published
.AddJsonFile("appsettings.json", optional: true)
.AddJsonFile($"appsettings.{env.EnvironmentName}.json", optional: true);
config.AddEnvironmentVariables();
})
Read more here: Sharing appsettings.json configuration files between projects in ASP.NET Core