The various claims of exotic groups probably have nothing to do with the “ten tribes.” Some of them are people in primitive areas who were converted by Christian missionaries, studies the Bible, and decided some of their pagan tribal customs were similar to what they read about (e.g. circumcision, taharas ha-mishpacha). Unless a group has distinctly middle eastern female DNA they are unlikely to be Jewish, but often consider converting.
To likely be of Jewish descent, one should look for separation of meat from milk, Shabbos, knowledge of which animals are forbidden – and most importantly, a credible explanation of how they get to a strange place. Knowledge of Hebrew helps. The Ethiopians pass. THe Lemba probably are related. The Bnei Menashe are a stretch. Groups in the Americas (other than possible descendants of marranos) or places such as New Guinea are clearly not related. Except for the Ethiopians, who were never really lost since there were contacts with Yemenite and Egyptian Jews over the years, all the others are highly unlikely to be Jews according to halacha.
Conversion is controversial since the Israeli government isn’t thrilled about converts who are non-white, and are likely to become strictly Orthodox (as opposed to secular East Europeans, who if “converted” will be good hilonim).