I am definitely not the most experienced person at doing this sort of thing, but it is important to make the distinction here that you are talking about responsive client-side behavior, not to be confused with Shiny's reactive behavior.
In any case, the responsive design stuff is typically done with CSS by Bootstrap or something like it. You can use Shiny Themes to inject your own CSS and force this behavior if you like. However, I would be curious to hear why you want this. Typically responsive design is more desirable than small screen users having to scroll, etc. Are you familiar with the Bootstrap framework?