His downfall is that he died. Probably around age 65. Various acounts suggest death from a battle wound, falling from a horse while hunting, or illness. I wouldn't hypothesise that he did anything that led to his "downfall" just that he died as relatively old warriors are prone to eventually do.
The British Empire was the largest by population, max land area and % of land. Although the Mongol area was very close on all three and had a higher percent of the total population of the world.
After Genghis Khan's death the Mongol empire continued to expand but the eventual power struggles between his successors ultimately led to the splitting and eventual fall of the Mongol empire.
There are a few other threads with more details about the Mongol empire in case you're interested: