The text says that the brother of Kim "did bring [Kim] into captivity; and he did remain in captivity all his days. Thus Kim probably stayed in the same local area from which he had ruled.
According to Glenn Scott, Kim became another of a long line of royal captives in the typical Asiatic pattern of his ancestors. Continuing in that pattern, while in captivity Kim begat a son whom he named Levi, who also remained a royal captive for forty-two years after his father's death. Whether Levi was released or escaped, Moroni does not say, but Levi organized a rebellion, overthrew his uncle, and reclaimed the throne. King Levi was one of several good rulers, and during his reign his people prospered. Moroni wrote that Levi "did live to a good old age and begat sons and daughters, and he also begat Corom, whom he anointed king in his stead" (Ether 10:16) (again the youngest son). King Corom ruled righteously and when he died his son, Kish, ruled as king. [Glenn A. Scott, Voices from the Dust: New Light on an Ancient American Record, p. 52]
Geographical [Theory Map]: Ether 10:16 - 10:32 City of Lib Built Where the Sea Divides the Land (Chronology)