I can't answer for all of North America, but I can provide a little perspective from the Southwest (Arizona/New Mexico, southern Utah/Colorado, and Sonoroa/Chihuaha). By the time Athabaskan people arrive in the area, the timing of which is still very much up for debate, around the end of the 13th century or during the 14th century AD, many of the areas they moved into which had previously been highly populated (such as the Four Corners region and the Arizona/New Mexico border region) were either already depopulated or in the process of being depopulated.

There are very few areas of the Southwest were we actually have evidence for Athabaskan people displacing the existing populations. Certainly there is very little evidence that they had any technological advantages over earlier arriving groups. As far as we know, bow and arrow are the primary weapons technology used by both groups. On top of that, it is difficult to get accurate estimates of nomadic population size, but it is likely that the sedentary agriculturalists already living in the Southwest had a much larger and denser population that would have been an advantage in warfare against the nomadic newcomers. The only cultural advantage the Athabaskan migrants would have would be the advantages conferred on anyone by living a primarily nomadic lifestyle, being the ability to utilize a wider range of environments and ecological niches than sedentary agriculturalists and so colonize many more marginal environments and locations.

That said, the primary advantage of nomadic groups in the Old World - the mobility provided by horses - was not a factor in warfare or other relationships in the Americas until after the Spanish introduction of horses into North America sometime around the 1520's to 1540's. This did massively change the dynamic between sedentary agricultural people and Athabaskan groups (the Apache in particular) but is more related to the nomadic lifestyle they share with non-Athabaskan groups like the Comanche or Shoshone, who also adopted horses and similar patterns of relationships with sedentary groups.