In particular, I am looking for the effects on China, Korea, and Japan. I understand the black death started in China, but I don't recall reading about the great mortality in anything Song China or the Kamakura Shogunate.
Was East Asia hit as hard as Europe, or were they less immunoniave to diseases of that nature?
Good question. In regards to China, we're not talking about the Song Dynasty, though. In the 1350s AD the Mongol-led Yuan Dynasty still ruled in Beijing, whereas the Southern Song was already conquered by 1279 AD. In either case, yes, there was a plague event in China that was partially responsible for the chaotic fall of the Yuan Dynasty and rise of the Red Turban Rebellion. A leader of this movement, Zhu Yuanzhang, later became the first emperor of the Ming Dynasty, i.e. the Hongwu Emperor, in 1368. Perhaps the effects of the plague weren't highlighted as much in China due to the fact that they were preceded by massive natural disasters and famines that killed millions of other people. Think of it like the "Spanish Flu" of 1918, the greatest pandemic of all time that killed millions, yet it was overshadowed in the public mind and memory by the casualties of the concurrent First World War. For many people these deaths were often conflated with the casualties of the war effort.
I'm not so sure about plague in Korea and Japan, but the Black Death and its effects in India and China are explored in the following article: