When using the approach to configure the WebPart for different webs, there is no inheritance and no global configuration. To achieve the same behavior, you should use the same configurations in all config files and/or resources.
When it comes to the CSS files used, you should note that they should be referenced in all MasterPages where the WebPart is loaded and additionally, to configure properly the exact config files according to the web scope used.
If you have further issues, I suggest providing the implemented config files, so that I could be able to understand better the situation.
Check out the Telerik Platform - the only platform that combines a rich set of UI tools with powerful cloud services to develop web, hybrid and native mobile apps.