[hellbender]

Genghis Khan, who led the greatest expansion of the Mongols, became Great Khan in 1206, during the Medieval Warm Period (950-1250 AD), not during a cold period.

[muawiyah]

The Mongol "build up" occurred over several centuries prior to Genghis, but his conquests were not a "volkswandering" sort of thing ~ more like an organized invasion by a major military power.

I think what the researchers focused on here were the internal Chinese disturbances ~ which took place even under Mongol control.