USSR (Russia) used to dominate world chess but that dominance seems to have faded away; with the Russians losing both the World Chess Championships and the Olympiad. What is it that the Russians are not doing right that the USSR used to do, and/or what is it that the other countries did to catch up and surpass?
Russia never dominated the world chess championships. The USSR did. Russia and the Union of Soviet Socialist Republics are two very different things.
The USSR consisted of -
Today these are 15 separate federations.
It is also instructive to look at where world champions have been born after 1948 and what their nationalities would be based on birth in today's world.
Mikhail Botvinnik (1948-57, 1958-60, 1961-63) born Kuokkala, Finland
Vasily Smyslov (1957-58) born Moscow, Russia
Mikhail Tal (1960-61) born Riga, Latvia
Tigran Petrosian (1963-69) born Tbilisi, Georgia
Boris Spassky (1969-72) born Leningrad, Russia
Bobby Fischer (1972-75) born Chicago, USA
Anatoly Karpov (1975-85) born Zlatoust, Russia
Garry Kasparov (1985-2000) born Baku, Azerbaijan
Vladimir Kramnik (2000-2007) born Tuapse, Krasnodar Krai, Russia
Anand Vishwanathan (2007-13) born Mayiladuthurai, India
Magnus Carlsen (2013-) born Tønsberg, Norway
So during the post war Soviet period the title was held by 3 Russian born players for a total of 14 years and by players from Finland (14 years), Georgia (6 years), Azerbaijan (4 years), USA 3 years and Latvia (1 year).
In the post Soviet period Russian born players have held the world championship for 7 years compared to Azerbaijan (11 years), India (6years) and Norway (6 years).
What is it that the Russians are not doing right that the USSR used to do?
Nothing. What the USSR did was to keep 15 nations trapped in a large communist prison, not allowing them to leave. This was not "right" it was evil. When the USSR collapsed and the prison walls pulled down (literally in the case of the Berlin Wall) the people who had been trapped were free to go to other countries to enjoy the benefits of living in democratic countries which respect human rights.
Top Soviet players ended up in countries like the USA, Israel, Spain, Holland, Germany, Switzerland, Ireland, etc. The list goes on. This obviously strengthened the chess in these countries and weakened the countries which were formerly part of the USSR.
The Soviet Union used to dominate chess because it had state-sponsored schools that provided high-quality trainers. After the Soviet Union broke up, funding dried up, and many of these trainers emigrated. Naturally the competitors caught up.
Vladimir Kramnik discusses some of this in an interview he gave with chess.com.
Chess.com: Much has been written about the legendary Soviet chess schools. You attended the Botvinnik school in the 1980s. What was it like?
Vladimir Kramnik: People always have a bit of an exaggerated picture of the Soviet chess schools, this idea that there were these concentration camps for kids, working 25 hours a day on chess. No, it was just a two-week session, held twice a year, and even during these two weeks, it was not so terribly intense.
The main asset of the Soviet chess school was actually the high standard of trainers throughout the country. In a world without internet and little exchange of information, this knowledge was hidden away behind the iron curtain. The reason the balance of power started to change after the iron curtain came down was because many of these Russian trainers emigrated, allowing western players and Chinese players to learn from them and get closer to the Russians.
You were 14 when the cold war ended and the Soviet Union began to collapse. How did that affect you?
Of course there were some issues, but I was just a kid at the time, so it didn’t make that big an impact on me. I think it was a much more difficult time for the older generations. For me, there were plusses and minuses. The plusses were that I could finally start going abroad to play tournaments, of my own will. The minuses were that all the state support for chess started to disappear.
The years 1990-1992 were a very difficult time financially in Russia, and many talented players of my generation didn’t manage to improve quickly enough. They ran out of money for trainers, and they ended up stuck at the same level. I was quite lucky in that I benefited from the state support of young talents when I was small, and then I reached the top of the game very quickly. But initially, money was not always easy.