They would not only have had to be very nomadic but able to adapt to the multitude of changing climates (each containing a different mix of new plant and animal life) as they travelled from North to South America. That is a pretty tall order.
As Physicist pointed out, that's only 10 miles/year. We don't know what kind of poulation pressure would have impelled them to keep moving as new people crossed over behind them. If we assume a coastal fishing people, they could move their villages large distances without trouble and not caring that much about minor changes in climate