The first project I got to work on that featured responsive web design was the Halo 4 Stats Web App. It was the first time I'd had to test the same exact app in the aspect ratio above and on a phone screen. It was a new thing to me, and required a new approach.
In consultation with stakeholders and the app devs, we decided a RITE test was the best way to approach the Halo 4 Stats Web App. RITE stand Rapid Iterative Testing and Evaluation, and is a method for quickly identifying issues and making changes. Instead of waiting until the end of a 6 person usability study to make changes, developers sat in the room with me making changes in real time as we observed issues. Users moved from device to device, using the same site in different ways, and we were able to make a huge amount of changes in a very short time.