Answer by Rory Young:
This is known as persistence hunting and I have done it. The most important evidence that humans persistence hunted in the past is the fact that we can do so today…The ability to long distance persistence hunt or even run really long distance is not a common one. The only African predators that do so (other than man) are African Wild Dogs (Lycaon pictus or Painted Wolves)
It is almost certain to have been used by early humans and is a theory for our having evolved the ability to run long distance.
I have hunted Impala in this manner some years ago. I noticed that when spooked they would not run far. I had also seen a tame Impala that sometimes walked with us on anti-poaching patrols had absolutely no stamina or endurance. Clearly they have evolved to quickly outrun predators and then recover.
One day I had to kill an impala for the pot and so instead of the usual bullet a game scout and I ran one down. It was not that difficult.
The San people of the Kalahari still persistence hunt to this day. They will often combine endurance running, tiring out an animal, with stalking after leaving it to settle once it is exhausted.