The above are links to the digging on Sergey Brin that I did mostly on Russian language sites.
The family was from Chernigiv now Ukraine and move to Saratov. This city was on the Volga river in the territory that used to be Khazaria and is not far from Kazakhstan's border. In the Soviet Union when someone did excellent work they didn't say "Well done". Instead they said "You'll go to Moscow". In fact that was the only way you could move to the capital so the fact the Sergey's grandad and dad lived in Moscow meant that they were part of the top tier of Soviet society.
So, why did they leave? Their story about Jews being restricted from education is fake, because clearly they did get an education. And the other half of the question is why did the Soviets let them leave? This was a brain drain and not something that you want to do in a controlled country. This makes the whole move seem suspiciously like an intelligence assignment abroad, and then they land in Maryland in the heart of the military and intelligence establishment.
But that is just suspicion for now. We will need some solid evidence to go further with this. I would like to find out more about the great grandma who became a Communist in the USA, lived in Chicago and then went back to the Soviet Union.