If there was a constant inflow of flies into the room, after some time you would have an infinite number of flies. I never heard of this happening!
One could argue that there is a constant number of flies per volume outside the room and when you kill all the files in the room, there is a negative gradient of flies and the insects will "naturally" move to the room. But such a "fly pressure" would require the flies to bounce from each other, like atoms, which does not seem to happen.
Or maybe flies are territorial animals?
Regarding territory, sort of: "In urban areas, where people live in close proximity and there is garbage, flies have small territories and may only fly 1,000 meters or so. Rural house flies roam far in search of manure, covering up to 7 miles over time." Via: https://www.thoughtco.com/fascinating-facts-about-house-flie...
With a constant finite inflow, you’d only have infinitely many after infinite time.
As to why they don’t keep amassing: flies need food, water (moisture), warmth and oxygen. Stuff too many in a room, and conditions elsewhere will be better for them.