Maybe pre-whiteman era but most if not all adapted to the new foods available. Read Lewis and Clark journals sometimes or history of certain tribes. Horse meat was used as well. You are right the hunter gathers of the plains did migrate with the game.
Not many American Indian tribes ate dog meat, chickened. Mainly those who were practicing some form of agriculture. The fully hunter-gatherer tribes did not keep animals for food; they hunted wild game, fish and fowl and were on the move seasonally to follow the meat.
My reference was to the "career" hunter-gatherer Native Americans, not the tribes that heavily practiced horticulture, agriculture and/or pastoral food raising for the bulk of their food. There is a different mindset and ethos between peoples who grow/raise their food, and those who are completely dependent on the availability of game and the necessity of truly knowing its nature.