The migration of Russian-Koreans from Kazakhstan to Uzbekistan started soon after their deportation in fall 1937. According to one Soviet document, the number of Russian-Korean households in Kazakhstan fell from 20,530 in late 1937 to 18,495 by 1 April 1939. This same document states that one of the two major reasons for this decline was the migration of significant numbers of Russian-Koreans to join relatives in Uzbekistan. The other reason for this decrease was the movement of Russian-Koreans from rural areas to towns and cities where they worked in enterprises and institutions and were not included in the 1 April 1939 count of Russian-Koreans resettled from the Soviet Far East to Kazakhstan (Li and Kim, doc. 66, p. 152). Unfortunately, from the limited data that I have, there is no way of knowing the relative weight of each of these causes in reducing the number of Russian-Koreans counted in Kazakhstan.
The movement of Russian-Koreans from Kazakhstan to Uzbekistan did not always occur at the instigation of those living in Kazakhstan. In some cases relatives living in Uzbekistan took active measures to bring their relatives from Kazakhstan to join them. One such Russian-Korean living in Uzbekistan was Nadezhda Kim. She worked on a silk sovkhoz (state farm) in the Khoja-Abad district of Uzbekistan, but her husband, brother's wife and three other relatives lived in the city of Karaganda, Kazakhstan. She wrote a letter on 10 February 1938 to the NKVD chief of her district asking for permission to travel to Karaganda and bring these relatives back to Khoja-Abad, Uzbekistan to live with her (document reproduced in V.D. Kim, p. 127). I do not know if she was allowed to travel to Karaganda or not. But, her letter certainly points to efforts by Russian-Koreans in Uzbekistan to bring their relatives from Kazakhstan to live with them.
V.D. Kim, ed., Pravda-polveka spustia (Tashkent: "Ozbekiston", 1999).
Li U He and Kim En Un, eds., Belaia kniga: O deportatsii koreiskogo naseleniia Rossii v 30-40x godakh (Moscow: "Interprask", 1992).