When the young boy is first being tested, he is presented with several pairs of objects to choose between. When choosing between two bowls, he picks up the bowl on the right and examines it. When he places the bowl back on the table, the bowl on the left is in his hand.
When the Dalai Lama flees Tibet, a blood blister is beneath his fingernail in extreme closeup shots, but not in medium shots.
For narrative purposes, the timeline is compressed; the Chinese invaded in 1950, the Dalai Lama visited Chairman Mao in Beijing in 1954, and he fled Tibet in 1959.
When the Chinese soldiers with goggles go marching by, the Caucasian-looking soldiers could be Uighurs, members of a Turkic ethnic group with Caucasian features. Most of them live in Xinjiang, just north of Tibet.
When Kundun is dreaming that he is surrounded by monks slaughtered by the Chinese, the monk closest to him moves his eyelids.
(at around 54 mins) The Dalai Lama asks Phalu if they can seek India's help. Phalu says India is a newly independent country, still struggling. In the next shot, it is 5 years later, 1949. That means that the previous shot was in 1944, while India was still under British rule. India got its independence on August 15, 1947.
When Kundun, disguised as a soldier, escapes Tibet in 1959, he walks through a Tibetan crowd that is singing "Tse-mey Yonten" ("Words of Truth"), a song composed in mid-1960.