NerdDinner uses Web.config for configuration—both for application configuration values that are constant between releases and for environmental configuration values that change between different environments. The configuration file is baked into the release package, which makes it awkward to change. In Chapter 3, Developing Dockerized .NET Framework and .NET Core Applications, I split the appSettings and the connectionStrings sections from Web.config into separate files; doing this lets me run a container with a different set of configurations, by attaching a volume containing different config files.
There are different types of configuration, though, and having to mount a volume is quite a heavy option ...