The introduction of horses by the Europeans significantly changed their cultures.

Various Sioux tribes were pushed west of the Mississippi around 1600-1700 by various wars that took place between natives at the time. It's probably the most well-known pre-contact migration of Native American tribes (at least those that live in US/Canada; the Mexica migration into the Central Mexico Valley is probably even more well-known, as it's a very key part of their own histories).

They are likely alluding to the trail of tears.

Neither the Sioux nor the Navajo were on the Trail of Tears, which affected the tribes who lived in the US Southeast, most notably the Cherokee, but also the Muskogee, Seminole, Chickasaw, and Choctaw.