I am an archaeologist that has had the lucky privilege of working in the United States, Canada, and Peru. As I am not a member of a dispossessed group myself, this is a very limited perspective but I hope to at least point people to sources and aspects of this topic to think about. I am also going to assume that the question refers to the past material culture of extant ethnic groups that were disenfranchised. I am also writing from the perspective of archaeology, obviously (^(please don't hurt me, historians)).

This is a very broad debate with very different attitudes, ethical considerations, legal reactions, and post-colonial contexts depending on the region and dispossessed group in question. There is considerable variation in the relationships between marginalized groups of people and research in material culture. Research can be seen as liberating, revealing and conserving a lost past that helps heal a fractured present and produces avenues of preserving and reclaiming traditions (ideally, through incorporating interpretations from descendants and breaking colonial barriers of archaeologist vs. subjects of study, etc.). However, research can also be seen as intrusive or, at worst, justifying and replicating colonial patterns of violence, alienation, and ethnocide. If you haven't picked up on it already, the colonial history of North and South America is unavoidable when discussing archaeology in these regions. This is because, well, that's how dispossessed people were "dispossessed" in the first place.

Examples:

Layla Renshaw's work on exhuming mass graves from the Spanish Civil War, or slave archaeology from the the Antebellum South as positive examples of recovering a "lost" or intentionally destroyed past.

Skull Wars, describes the Kennewick Man controversy and details the history of deteriorating relationships between scientific archaeologists and Native Americans in the United States. Much of this animosity and distrust stems from the colonial history of archaeology in which anthropology was generally part of colonial systems and involved the collection (i.e. theft), analysis, and display of human remains as scientific curiosities. Some of the biggest museums in North America, such as the Smithsonian, still retain many of these skulls and have ongoing repatriation problems.

Methods:

Methods can include ethnographic data from descendants in combination with historical methods and processual and/or postprocessual archaeological fieldwork. The bioarchaeology of marginalized populations is also an interesting area of research for this topic.

A large part of the debate of how to ethically research dispossessed people surrounds who owns the data and material culture itself. Access, ownership, and control of information can all spark different controversies, especially concerning who gets to benefit from research activities. There are many data sharing and intellectual property issues being recognized and discussed by archaeologists, heritage practitioners, descendant communities, and policy makers. Questions that get asked: Why are we doing this research? Who gets to make interpretations of the material culture? Who is this research for? Who gets to read, see, and use this data?

If you're wondering why there is even a divide between "archaeologists" and "dispossessed groups" of people, reading about indigenous archaeology might be of interest. American archaeologists are overwhelmingly white, so indigenous vs. settler politics heavily affects North American archaeology. Basically, institutionalized racism and the colonial history of antagonism in the United States and Canada between native groups and researchers has made indigenous archaeology challenging to implement (to say the least), but there are examples of a transition into a more integrated approach to recovering and interpreting material culture. Theoretically and ethically, current trends seem to be shifting towards "community based cultural heritage research," in which descendant communities are directly involved. The goal of such an approach is to involve a community with all aspects of the research process, from project design to authorship. There has also been an increased interest in using digital technology to make this possible (but getting into that too much might digress from the topic).

More on theory, practice, and ethics from a Canadian perspective: https://www.sfu.ca/ipinch/

Note: I have known quite a few archaeologists that would resent the perspective I am writing about and view these developments as endangering objective research. While I theoretically disagree with such viewpoints, it is important to recognize that there is a lot of diversity of thought around how researchers should approach studying the past populations of North and South America.

Some examples of collaborative projects (that use digital technology to facilitate communication and reconstruct and conserve objects) between archaeologists and native groups that acknowledge or attempt to address these issues:

• Haukaas, Colleen, and Lisa M. Hodgetts. 2016. "The Untapped Potential of Low-Cost Photogrammetry in Community-Based Archaeology: A Case Study From Banks Island, Arctic Canada". Journal of Community Archaeology & Heritage. 3 (1): 40-56.

• Hennessy, Kate, et al. n.d. The Inuvialuit Living History Project: Digital Return as the Forging of Relationships Between Institutions, People, and Data. Museum Anthropology Review; Vol 7, No 1-2 (2013): After the Return: Digital Repatriation and the Circulation of Indigenous Knowledge; pp. 44-73.

• Lyons, Natasha, David M Schaepe, Kate Hennessy, Michael Blake, Clarence Pennier, John R Welch, Kyle McIntosh, et al. 2016. "Sharing deep history as digital knowledge: An ontology of the Sq’éwlets website project". Journal of Social Archaeology. 16 (3): 359-384.

• Ngata, Wayne, Hera Ngata-Gibson, and Amiria Manutahi Salmond. 2012. "Te Ataakura: digital taonga and cultural innovation". Journal of Material Culture. 17 (3): 229-244.