What roles do anthropologists play in migration?

It is helpful to clarify the terms you are using in this question to understand the connections between migration and anthropology.

"Cultural Anthropology" as the social-scientific field we know it to be (an epistemological methodology, i.e., a method of inquiry used to understand or create knowledge) emerged from the Enlightenment period of the 1700s in Europe. During this time, primarily wealthy and educated individuals began pursuing science and philosophy, and anthropology as we know it today came out of this interest.

This coincided with the colonization of Africa and the Americas. As leaders and educated classes from European nation-states went to civilizations in Africa and the Americas, they often brought with them explorers (and, later, anthropologists) to "study" the people and societies of these places. Early anthropological work was produced in tandem with colonization: for instance, missionaries and explorers would take ethnographic notes on the people found in these places that were considered very "other" when compared to Europe. Early anthropology has been critiqued in modern academia as a study of the other.

"Migration" is also a historically specific phenomena. It can be understood to mean several things: immigration, when individuals or groups of people leave their countries of origin to go to other countries for various reasons, including family, job opportunities, medical needs, or education needs, among other reasons (if a person immigrates due to a political or natural occurrence, this person might be considered a refugee in the new country); asylum seeking, or leaving a country of origin due to discrimination and persecution; and settler colonialism, such as what happened in the United States with Europeans settling and taking over land previously lived in and governed by indigenous people.

With these contexts, we can understand the development of the field of cultural anthropology as directly connected to settler colonialism—as Europeans moved to the Americas and Africa and took over governance, so too did they create work studying the "other."

Currently, anthropologists work all over the world, and though this context is still present, individuals with connections to nations that have been colonized have utilized the field of anthropology to write ethnographies that do not "other" their subjects, like Palestinian American anthropologist Dr. Lila Abu-Lughod.

Additionally, many anthropologists from all over the world study phenomenons of migration.

Anthropologists are no more or less likely to be migrants or play a role in migrations than members of any other profession. Some anthropologists do travel for their work, but, as all forms of business travel, this is not considered migration.

Anthropologists do study migrants, including both internal migrants, migrants who move from one country to another, and migratory tribes of hunter-gatherers or nomads who have no fixed settlements.

There are two main areas of interest to anthropologists in the study of migration. One is studying immigrant communities and the other is studying the process of migration itself.

A recent area of interest of scholars in cultural studies has been diaspora studies, looking at how a particular diaspora (e.g. Indian, Tamil, Jewish) is affected by the locales to which the group has migrated.

What role do anthropologists play in migration? Give three examples.

Rather than play a role in the act of migration, anthropologists consider and analyze evidence in order to gain a better understanding of the way in which migration has occurred throughout history. By doing so, they are able to track the migratory movements of different peoples and cultures and develop theories which might explain why a particular population migrated from one geographical location to another at a particular time in the past. In this way, they might investigate migration after it has already occurred.

For example, anthropologists study early human migrations out of and into Africa. Since many migration waves are merely reported or theorized rather than proven, anthropologists may use the evidence found at archaeological sites in an attempt to develop a holistic picture of when, why, and where migrations occurred. Understanding this information will ultimately lead to a better understanding of our history as a species.

As a second example, one of the largest human migrations in history was that of the African Americans from several southern states in the US to several states in the north. This migration occurred between the years of 1910 and 1970 and led to significant cultural changes. Why and how did this migration occur, and what was the nature of those cultural changes? These are just a few of the questions that an anthropologist may investigate.

Lastly, many settlers and Puritans migrated from England to Massachusetts and Barbados between the years of 1620 and 1640, purportedly compelled by fear and the desire to seek religious freedom. This is referred to as the Puritan migration, and it may be studied by anthropologists in order to better understand the nature of the interface between religion and geographical migration.

The largest role that Anthropologists would play in migration would have to be in what is termed "Forced Migration".  Some examples are:

Removal of 8000 Israeli citizens from the Gaza Strip in 2005.  Because of conflict over the land, that dates back thousands of years and recent escalation of violence, the Israeli government decided to relocate the 8000 or so Israeli citizens living in the area.  Homes, businesses and factories were all demolished and the people were relocated (most to Negev).

Relocation of 1.25 million Chinese residents for the building of the Three Rivers Dam in the Yangtze River.  Another estimated 2 to 4 million people relocated towards urban areas after the dam was built.  Hundreds of archaeological sites were destroyed.

Varosha, Cyprus, in 2005 when the entire population of approximately 35,000 people fled because of the history of the invading Turkish Army.