Their lands were not bought. The government forced them to trade for wilderness land given to the group and took land owned by individuals. They looted the homes before burning them to the ground. Then the government gave the land to Whites by lottery.
The Cherokee were part of the Indian clearance in the South, but there was also an Indian clearance in the Northern states. Buley covered the big payout in Chicago. The southern clearance was less organized, but payouts occurred.
One of the reason for the Dawes Rolls was to enable the Cherokee, as they organized themselves, to DISQUALIFY what turned out to be 25,000 "others" who appeared to follow them to Oklahoma ~ a large number of whom were grifters intent on stealing Cherokee proceeds from herd and other property sales.
Not everything happened in all places the same way.
The whole idea of moving everyone West was the doing of Thomas Jefferson and the Kickapoo Indians anyway.
Without knowing the germ theory of disease they deduced that Indians living in the presence of whites and blacks were dying at alarming rates so any chance at biological survival required relocation.
You might look up John Metoxen (I believe he's a relative of mine ~ his sister is identified as "Ox In" married to an ancestor). He was a Brotherton, and I suppose technically adopted by the Oneida at the time of the conquest of the Mohicans, and he became quite a challenge to Jefferson and crowd. John advocated removal ~ and all he wanted was Indiana Territory ~ later briefed up to Wisconsin, and for a while he bought into the idea of moving to Oklahoma.
The Cherokee had unfortunately set up shop in an area where Jackson's friends wanted the land. He was less couth than Jefferson.
Still, the folks who walked were the black slaves. The Indians left on horseback and with wagons. In places they actually rode some of the earliest trains in America.