My understanding of enroute is you're not really working multiple airports, you're working at a center.
Ie: ARTCC, an air route traffic control center, typically handles instrument traffic at the flight levels (18k ft and up for most commercial traffic, albeit occasionally below). Traffic may literally just be passing through your Flight Information Region, and not landing, albeit they could be landing. If they are, you typically would hand off to a terminal facility.
en.m.wikipedia.org
This is from what I know from being a pilot already, not from working this job, please correct me if I'm wrong current ATCSes. ?