Sites can vary, but most areas you can stay in a specific spot for 14 days then move to another spot in the same or a different forest for another 14 days. Note you reallly do need to read about the specific forest and the specific area. Some allow only 7 days or 10.
Usually you have to move at least a certain number of air miles from your current camp after you exceed the stay limit to continue camping on public land. The distance varies by jurisdiction, on BLM land in Arizona it's 25 miles, Colorado national forests require 30 miles seperation. There's also a time limit before you can return to an area, usually 14 days.
The idea is to prevent people from semi-permantly residing in an area by just moving a short distance and then returning.
Whether you move from one forest to another is irrelevant.
While there is a 14 day limit in Colorado, that rule is suspended during certain times of the year. ie: I go to the White River National Forrest every year for hunting season. I usually get there the first week of October and don't leave until around November 10. I see the wardens several times during the stay and they never say anything to any of us about leaving. It might be that since we are out of state hunters we are spending thousands of dollars to be there so they just leave us alone. However, there are big signs at the entrance of the camp area saying the 14 day limit is strictly enforced.
The best answer is to ask the Ranger what he would like.
(you are supposed to check in with them when you arive).. If you don't see a Ranger it does not mean they don't see you, so don't lie about when you got there, as they sit on mountain tops with binoculars.
Sometimes all they want is for you to go out of the BLM zone and come back in. I think they just don't want you to start making 'camping improvements' to the BLM lands .