In Tableau desktop, the size of the dashboards should be set to "Automatic" before you publish to Server/online. When this is done visualizations will automatically resize if properly embedded.
Thanks for responding to the question. The answer is still not the right one. A responsive design would lay out the various title vertically and provide you with a "hamburger" menu. All automatic does is stretch the layout. What should take place is that all the charts should come one on top on the other in a vertical manner. Look at any of the public visualizations, none give you that popup menu. Is this a limitation of the product?
This clarifies the question.
I would tend towards saying yes it is a limitation. There might be some complex workarounds out there to do what you describe, but I'm certainly not aware of any native functionality to do this.
Mind you, I'm not aware of any other visual analytics dashboard programs that does the kind of responsive design you are talking about. This would certainly be an interesting feature.