I've always had a bit of a problem with responsive design. It too easy to assume the most important context is the size of the screen, too easy to fall into the habit that the way you build a mobile version of a service is to change the presentation layer - just shuffle the same content about the page in a different order and hide a couple of things *.
To make things worse, the term 'mobile-first' is often understood by non-developers as "we designed this product for a mobile first!" rather than "we use the mobile CSS as the base and build up from there".
Websites that work on a mobile are not the same as websites designed for a mobile context. Resizing a browser to make sure it looks OK is probably not good enough any more:
Some tasks are just better suited to mobile/touch or desktop/keyboard - maybe the desktop and mobile version of the service are fundamentally different propositions? It's worth taking a look at what Evernote have done as a result of understanding the context of how the mobile (app), desktop web and desktop app versions of their product are used. The GOV.UK performance platform team are also exploring this, by making the big screen view of data fundamentally different. Knowing when to build one product or multiple products is going to become increasingly important (I think).
The design language for mobile software is diverging from that of bigger screen software (remember in the late 90's early 2000's when lots of websites looked like desktop software with loads of dropdowns and side menus?). So long as all mobile web apps look like mini-me versions of the desktop browser one (it is hard to find many that do not), they are at risk of being beaten by native apps. Users just are going to start expecting better.
Mobile phones and tablets, with touch as the interface, are set to become the default way people consume the web so we better make sure the mobile web is as good as native apps, or there's a good chance the web will lose.
I think it's time for teams to start coding directly on mobile (or at least an emulator)* and for product owners to start demanding demos on mobile in the first.*\* I'm not suggesting everyone does this, just that is an easy trap to fall in to.
** I'd be interested to know if anyone has seen any good development setups that replace coding in browser on the development machine with coding on mobile.