If you volunteer recycling trucks to another city, does your city have to process the recyclables?

In one of my regions, I have a large city surrounded and support by several smaller cities. Many of which do not have recycling centers, but have an educated enough populace to start recycling. I have volunteered a few vehicles around the region to pick up their recycling and reduce their garbage load. What I want to know is...

Is the recycling that I am picking up in other cities actually coming back to my city for processing?

If it is, it seems like a great way to make a profit by collecting more recyclables, and then selling the plastic, metal, and alloy on the market. However, the recycling center does not have much storage, so it would seem that it would fill up pretty quickly if it were actually taking in all this stuff.