There seem to be three major ways to claim portions of the past as one's heritage -- genetic, cultural, and geographic. So my ancestors could be variously defined to include the Vikings, the ancient Greeks, or the Lenape/Oneidas/Nipmuc(depending on whether I count Palmerton, Hamilton, or Worcester as my residence).

Disputes between Native Americans* and archaeologists over the use of human remains can be seen in terms of these types of claims to heritage. In the case of very old bones, such as Kennewick Man, the Native American argument is geographic (ideally all three types of connection can be shown). Native groups often treat geography as the fundamental component, claiming continuity with all people who lived on their land before them. From this they can infer genetic and cultural continuity, based on the claim of indigenousness -- that they are the true people linked to that land since time immemmorial. This is reinforced by the idea that deceased members of the tribe are united with the land, fusing the two kinds of continuity. The claim by archaeologists that most disturbs many Native activists is that Native tribes migrated to their contact-era lands (though such migrations form a part of much Indian traditional history and mythology), best captured by the Bering Strait migration theory -- a claim that undercuts total geographic continuity.

However, geography only works up until 1492. After contact, it is obvious that Native Americans' lands have also been inhabited by those whose claim to remains is being disputed. By purely geographic standards, that would give white archaeologists a claim (while making the claims of forcibly relocated Native tribes more tenuous). So the primary line of heritage from contact to the present is made genetic. Meanwhile, efforts are made to portray non-Natives as illegitimate and visitors to the land they happen to reside in (a claim made easier by the reluctance of European culture to acknowledge connection to the land in a way similar to its articulation by Indians, as well as the effect of the wilderness myth in making America out to be a blank slate at the time of European colonization).

Counter-arguments to Native claims usually attack a genetic or cultural lineage. Archaeologists will argue, for example, that the remains in question are not biologically ancestral to the Native Americans trying to claim them. Or they argue that that the culture of the deceased was so different from that of the claimants that they can't reasonably be considered the same group and Native beliefs (about things like proper burial practice) are thus not relevant. The problem here is that archaeologists are asserting a prerogative, not recognized by the Natives, to define "how close is close enough" for these types of relationships. There's no clear line between "ancestral" and "not ancestral" on any of the three dimensions of heritage, which means there's wide room for disagreement even if neither side is drawing the lines based on self-interest.

From a Native perspective, (white) archaeologists can come off as making a geographic claim to remains -- "this is our land now (by right of conquest), and thus we can do with these remains as we please." However, the archaeologists themselves see their claim as transcending the three categories I outlined. These three categories are types of claims to particular heritage -- for one person or group to claim a special relationship to another person or group. Archaeologists, however, see their claim as on behalf of the whole of humanity (genetic insofar as they would dispute the ownership of the remains with, say, a hungry dog).

*nb: My references to "Native Americans" should generally be read as a shorthand for the subset of Native Americans who promote a separate Native cultural identity and more or less agree with the positions I attribute to them. These people often claim to speak for Native Americans as a whole.