Before the Mongols conquered Song China, the first conquered the xixia, the entire central asia, persia, the abbasid caliphate in Iraq, and then Russia. Their empire was then ten times bigger than China. The mongols then drew upon soldiers and engineers from all parts of their massive new empire and it took them over 30 years to conquer Song China while they swept over the steppes of Russia, central asia and to baghdad in a few years. The Mongols also used hundreds of thousands of northern Chinese soldiers from the former Jin dynasty to conquer the Song.
The Manchus first defeated the Mongols and Koreans and forced both of them to ally with their state. They then drew thousands of defectors from China. The manchus could not even penetrate Shanhai pass at the great wall until the Ming general Wu Sangui defected to their side with his army, because rebels under Li Zicheng overthrew the Ming emperor and killed Wu's father. They then managed to take over China only with the help of Wu Sangui and his massive army of .ming defectors. In fact most of their soldiers were not Manchu but Han people.
Britain ruled over a massive empire already when it attacked China in the Opium Wars. Britain ruled Canada, India, singapore and parts of Malaya and colonies in Africa. The British used armies of Indian sepoys to attack China and fleets based in India and resupplied in Malaya and Singapore. Te population of their entire empire was near China's population and their territory was bigger than China.
All those empires which attacked China were bigger in China in botth population and area.
Population of Mongolia was much less than population of myth China, Mongolian were on horseback riding around the world, they ruled you only.
and Man Chus (Jerchens), they were also with very few number people.
and Japan ? read more about first Sino-Japan war.