I think I addressed this in another thread but here goes from my experience. We ran 100 free range layers with the cattle in on place. Fly populations were definitely less as they really worked on the cow patties. Ran chickens at another place about 3 days behind the cattle. The cow piles that the flies lay their eggs in have to "cure a bit" so I've been told, guess they need a bit of a crust but no one has ever explained it. But the 3 days behind allowed the flies to lay eggs and the larvae to hatch and they cut down on the fly population more than the ones that ran continuously. Thing is the chickens have to be active and really scratch the piles. Some breeds do better than others, meat type don't do nearly as good, but young growing birds and some breeds of layers do pretty good. The thing is to look up the life cycle of the fly that you have most common and figure that the chickens have to be working on the cow patties before the larvae develop and turn into flies from the larvae stage. I think the 2-3 days behind is kinda a standard to go by. It helps also if the pasture grass is short enough that the birds can really cover a certain area without getting lost in the "weeds" so to speak.
Having done both, I think the rotating may work a bit better, but our biggest problem is/was/ will be predators.