In the movie, when Kundun is disguised as a soldier escaping Tibet in 1959, His Holiness walks through a Tibetan crowd that is singing the song, "Tse-mey Yonten" (Tibetan name) also known as "Words of Truth". However, in actuality, this song wasn't composed until mid 1960, a year after Kundun had escaped to India in exile.
(at around 54 mins) The Dalai Lama asks Phalu if they can seek India's help, and Phalu says that India is a new independent country still struggling. In the next shot, it is 5 years later, 1949. This means that the previous shot took place in 1944, while India was still under British rule. India got its independence on August 15, 1947, three years after Dalai Lama asks for India's help.
When the young boy is first being tested, he is presented with several pairs of objects to choose between. When it comes to choosing between two bowls, he picks up the bowl on the right and examines it. When he places the bowl back on the table, he has the bowl on the left in his hand.