I would agree with the previous answers but I would like to add that once the native children were ripped from their families and sent to Indian schools, it was the final destruction of native culture and oral traditions. In the native societies, family was important, children were important. To take these children away from their families and teach them that their culture was evil was a sin. These children were punished for speaking their own languages and practicing their own culture. When they did return to their families, they had been "assimilated". Their language and culture had be beaten out of them. What better way to destroy a people than by brain washing the children of that culture? It is true that in some cases, native languages and culture were preserved but a high price was paid.
I would argue that the assimilation of Native American tribes began long before the Dawes Act or even the Indian Removal Act. Rather, the oral literary traditions of these peoples were certainly impacted in a negative manner almost from the first appearance of Europeans on the North American continent. The presence of Europeans, their culture, and even the tools and materials they used slowly began to erode the fabric of the culture and way of life of native tribes. Because of this, their oral traditions were impacted as well. As the imperialism of Europeans continued, this situation only grew worse. By the time of the Civilization Fund Act of 1819, it became official government policy to "civilize" and assimilate Native peoples.
There is no one size fits all answer for this question. Assimilation, remember, was forced in the form of the Dawes Severalty Act, which created a reservation system and required all Native youth to attend boarding school to learn English and Christianity. While this didn't necessarily change the oral tradition itself, it did change the ability of the tribal elders to pass on their history to the younger generation. Many of the languages died out within two generations, and much of the history was lost with it. While the tradition of storytelling and the passing on of family stories remained, one could argue it was permanently damaged by forcible assimilation.