The migration of persons from and to Russia due to the war is hard to estimate, but there is consensus that very many have left to former parts of the Soviet Union, while many Russians who had settled in Ukraine during the Soviet period have returned to Russia.
There have been two waves: the first immediately after the war began; the second, after the new draft announced on September 21.
The single best analysis of the first wave was published in English by Riddle on July 25:
A team of sociologists from OK Russians conducted several surveys on Russian emigration and found that the majority of migrants settled in Turkey (24.9%), Georgia (23.4%) and Armenia (15.1%). Following the outbreak of the war, Israel greatly simplified its repatriation rules for Ukrainian and Russian Jews in March 2022. About 2.8% of respondents moved to Israel. Slightly smaller numbers of migrants arrived in Uzbekistan, Kyrgyzstan and Kazakhstan — about 2% of all respondents. The most popular European destination countries were Serbia (1.9%), Montenegro (1.7%), Estonia (1.6%), Germany (1.6%) and Spain (1.5%) (see Figure 1). The choice of countries was largely random and depended on the availability of tickets and the rules for entry and stay. This was noted by 58% of respondents. Less than half of the sample plan to stay in these countries (43%). A quarter plan to move on (18%), 35% felt confused at the time of the survey, and as few as 3% plan to return to Russia.
Pre-war migration patterns were complex due to the demographic of the Soviety Union.
As of 2020, Russia had 11 million foreign born persons, 3.2 million of whom came from Ukraine (which was about half of all Ukrainians living outside the country), and 2.6 million from Kazakhstan. (Go here.)
An expert on area migration said this in May 2022: “According to our estimates, from the late 1980s to 2017 inclusive, there are about three million people who were born in Russia and live in far abroad countries. That is, not 11 million [as in the UN data], but three. So, if you use UN statistics, you should, if possible, remove the former Soviet republics from it. That will be more correct. For example, many people were born in Russia and moved to Ukraine during the Soviet era. Or take the “punished” peoples: Latvians and Lithuanians returned from exile with children who were born in Russia.”
Western estimates of emigration due to the war are extremely tentative.
The Carnegie Endowment says that 100,000 people moved from Russia to Kazakhstan immediately after the invasion.
The Economist estimates for the first wave that “A survey of recent émigrés showed that about a quarter of those who left settled in Georgia. The same number went to Istanbul, and another 15% ended up in Armenia. (These places do not require visas for Russian citizens.) Nearly half of those who left work in computing and it, according to a survey carried out in March. Another 16% were senior managers; 16% more worked in the arts and culture.” Note that these figures are from early 2022.
The New York Times estimates that, for the second wave, 200,000 have left since the new draft began in early September.