There have been cases of white people becoming parts of native tribes in the 18th century, however, I could not find anything after that. The best possible answer would address all races. Thank you!
Elijah Nicholas Wilson was a young boy when he ran away from home to live with the Indians. He became the adopted brother of Chief Washakie, who enticed Wilson to join him with the gift of a paint pony and a promise to take him away from daily chores of a homesteader life.
If I recall correctly Wakashie's younger brother died of sickness. His grieving mother wanted another child to replace her lost son so Wilson was chosen. In Wilson's biography "The White Indian Boy: The Story of Uncle Nick Among the Shoshones" he tells us that Wakashie made every effort to ensure that Wilson was aware of the consequences of leaving his family and joined them on his own free will. During his time with the Indians Wilson always had the option to return to his home and family but, fearing reprisals, the Indians kept him hidden from sight whenever they were close to white traders or settlements
Wilson was born in 1842. He ran away from home when he was 10 years old. His memoir helps give us an insight into the lives of the Shoshone people during the middle 1800's and is a good read today.