Sounds as if that shift may have occurred at the end of the Younger Dryas, with changed climate conditions.
The last I read about it, that savannah had been grazed continuously for 14,000 years prior, long enough to have endured climate cycles without interruption. Together with the change in albedo resulting from abandonment and an increasingly fungal soil is why I think the anthropogenic attribution as a cause of desertification may have validity.
Climate changed, no doubt, but it may be as an effect, a cause, or both.