If you develop your website as a shiny application, you will have a server with R running code, and you can invalidate with any mechanism you want I think. There is already a function to cache plot
I am not sure if you mean that, but child document is sometime used with R markdown about the code source to be in several document. One main Rmd and some child, to be merged together at rendering time. For html static website, this is not the same. I would break the site in several html and link that using html code and js in a main page. There is also the rmarkdown website option
Flexdashboard also have a self_contained option. They are based on the same kind of document as html_document. It is the template and them that changes. There is a self_contained argument. You can also insert shiny app element
Look also at this documentation to get details