When I look at maps of North America prior to the 16th century they often depict areas which are controlled by certain tribes. These are often very large areas, sometimes larger than modern States. How much central control did the tribal leaders have over these areas? Was anyone allowed to live within these borders, or did the tribal leaders/councils control who was/was not allowed in? Also, was the concept of territorial control (static borders, control of resources, etc...) common among the people of these tribes?
I realize this question covers a very broad area but for the sake of this discussion I will define the geographical area of my question as stretching from the Northern border of modern day Mexico to the Southern borders of the Territories of Canada (Yukon, Northwest Territories, Nunavut). Everything between the Atlantic and the Pacific oceans. The time period I'm interested in is ~1000-1491 c.e.
As you suggested, this question has multiple answers depending on both time and region. Even within the Southwest specifically, there's a great diversity in effective control. Let's start with the Hia C-ed O'odham, who inhabited the desert between the Gulf of Mexico and the Southwestern limits of Hohokam territories along the Gila river tributes. Within this large area of people there was a continuum of lifestyles ranging from the entirely nomadic Pinacateños to the semi-nomadic Areneños to the sedentary Quechan (who are generally considered a distinct culture). The Pinacateños are named for their territory, a vast swathe of desert covered in black lava fields around the Sierra Pinacate. These inhospitable lava flows have an extremely well defined area. The Pinacateños had a reputation for hostility towards outsiders. It's telling that inside of this large area, only two sites on the very fringes of the flows have petroglyphs (a behavior the Pinacateños never adopted from their northern cousins), suggesting the interior fields were largely closed to outsiders.^1
Moving forward and East 200 years and 300 miles, we can find Chiricahua Apache territory. In the early years of Apache occupation, the Apache were not a dominant power. Their territory was nominally under the control of older tribes and the Apache used hidden caches to keep their enemies from discovering emergency resources. As the Apache consolidated control over their territory, they gradually pushed out the other tribes.
However, the Chiricahua tribe was not a single political entity. In actuality, this name refers rather to their cooperation during the Apache wars to fight the US, Mexico and Spain after disease had devastated their numbers. In actuality the tribe was composed of a number of distinct, autonomous bands that inhabited a well-defined piece of territory within the ill-defined "Chiricahua Apache region". Families were spread across multiple bands^2 and so some degree territorial boundaries could be bent. On the whole, there was no overarching centralized control over the entire territory, but a chief would have great control over the well-defined and fluid boundaries their band established.
In general, conceptions of territory varied wildly. In some cases, it was so thin as to be an entirely useless concept while in others, possession was rigidly defined and policed. Even within a single cultural group, territory might mean different things at different levels.
^1 J. Hayden is generally the source on anything involving Pinacate, but his stuff is a royal pain in the arse to cite because so little of it has been digitized for look-up.
^2 This is a feature specific to Western Apache and most strongly expressed in the Chiricahua. The situation for the Eastern bands was generally closer to bands being distinct familial units.