Outright removal was one method, which was even sanctified by law and national policy in the 1830s and beyond. But that policy only resulted in relocating the tribes to new lands that the dominant society would want in the future.
So, new methods were adopted in a twofold strategy to disappear the tribal nations: the first was for physical attrition that would be through warfare, pestilence, starvation, benign neglect, and interbreeding with whites. This would kill off the people or force them off their lands in search of sustenance and livelihood.
The second was for cultural attrition that would be accomplished by “civilizing” the Indian people through Christianization, education, and assimilation. The tribes, it was hoped, would eventually abandon their savage ways and become brown white people, so to speak. Most importantly, they would become capitalists and recognize the importance of making the greatest use of their lands, mostly by ceding or selling them to the superior and dominant European Americans.
For examples of this myth, see Europeans Keep Discovering Newfoundland and Vanished Mound Builders in The Prairies. For more on what the Europeans believed, see Conservative Christians Are Oxymorons and Those Evil Europeans.