hi @nmclain and welcome. It is all about how you use to work. I would like to mention that coding is the “easiest” part when you know what to do. What I mean is that most important thing is to spend time on preparation eg. great job on not only on design it self (desktop and at least mobile too) but have elements structure in mind. This will prevent to solve major problems beforehand. I have worked with designers that deliver only desktop version without even thinking how it will look from UI/UX point of view on mobile. These things were later consulted, revisited and changed. So preparation is king.
When preparation is done and you are ready to move it from paper to code I work between all viewports as I need but generally when I work on one page I do rough structure of whole page but when is come to detail and responsiveness I will take one section and make it work across viewports. Then I will move on other page section. In meantime checking that everything is working across all major browsers. And so on and on and on. 
But this is only very short overview of my basic workflow and it can change from project to project as creating page structure and styles are not only things involved when working on project.
m2c