I obviously can't give one answer for all groups at all times, but historically the correct answers have often been 'yes' and 'no'.
In the 1600s and 1700s, there was a significant tendency for white people to escape the colonies and join indigenous groups. According to "Lies My Teacher Told Me" by Loewen, some colonies or forts set up guards not to keep the natives out but to keep the white people from escaping to them.
Side note: I highly recommend "Lies My Teacher Told Me", which goes through 5-10 of the most common grade school history text books topic by topic and analyzes their propaganda value: what omissions, errors, unspoken assumptions, etc create the narratives they describe in different parts of American and European history which give our kids a false understanding of their heritage? It was unbelievably enlightening.
For an online reading about this 'white flight' phenomenon, here's a quote from "Forgotten Founders"; "Franklin" is actually Ben Franklin writing about the indigenous in the colonies:
As they sought a middle ground between the corrupting overcivilization of Europe and the simplicity of the state of nature in which they believed that many Indians lived, Franklin and other Deists paid abundant attention to the political organization of the Indians, especially the Iroquois, who were not only the best organized Indian polity with which British Americans had contact, but who were also allied with them. "Franklin had the conception of an original, pre-political state of nature in which men were absolutely free and equal -- a condition he thought admirably illustrated among the American Indians," Eiselen wrote in Franklin's Political Theories (1928). Franklin himself wrote: "Their wants . . . [are] supplied by the spontaneous Productions of Nature" and that they did not at all want to be "civilized."
This state of nature was eagerly sought by many eighteenth-century Euro-Americans. To understand how many Europeans left their own cultures to live with the Indians is to realize just how permeable the frontier was. To those who remained behind, it was often rumored that those who had gone over to the Indians had been "captured." While some captives were taken, more often the whites took up Indian life without compulsion. As Franklin wrote to Peter Collinson May 9, 1753:
The proneness of human Nature to a life of ease, of freedom from care and labour appear strongly in the heretofore little success that has attended every attempt to civilize our American Indians. . . . They visit us frequently and see the advantages that Arts, Science and compact Society procure us; they are not deficient in natural understanding and yet they have never strewn any inclination to change their manner of life for ours, or to learn any of our Arts.While Indians did not seem to have much inclination to exchange their culture for the Euro-American, many Euro-Americans appeared more than willing to become Indians at this time:
When an Indian child has been brought up among us, taught our language and habituated to our customs, yet if he goes to see his relations and makes one Indian Ramble with them, there is no perswading him ever to return. And that this is not natural [only to Indians], but as men, is plain from this, that when white persons of either sex have been taken prisoners young by the Indians, and lived awhile among them, tho' ransomed by their Friends, and treated with all imaginable tenderness to prevail with them to stay among the English, yet within a Short time they become disgusted with our manner of Life, and the care and pains that are necessary to support it, and take the first good Opportunity of escaping again into the Woods, from whence there is no reclaiming them.Franklin followed with an example. He had heard of a person who had been "reclaimed" from the Indians and returned to a sizable estate. Tired of the care needed to maintain such a style of life, he had turned it over to his younger brother and, taking only a rifle and a matchcoat, "took his way again to the Wilderness." Franklin used this story to illustrate his point that "No European who has tasted Savage Life can afterwards bear to live in our societies." Such societies, wrote Franklin, provided their members with greater opportunities for happiness than European cultures. Continuing, he said:
The Care and Labour of providing for Artificial and fashionable Wants, the sight of so many Rich wallowing in superfluous plenty, whereby so many are kept poor and distress'd for Want, the Insolence of Office . . . the restraints of Custom, all contrive to disgust them with what we call civil Society.
