Has there ever been a Soviet or Russian equivalent? Even if its scope or findings were timid or muted in comparison? Especially in the aftermath of Glasnost?
The Soviet Union had the Central Committee and the Politburo. The Central Committee of the Communist Party of the Soviet Union was the executive leadership of the USSR while the Politburo was a political bureau and the highest policy-making authority in the Communist Party that was founded in October of 1917 and people in the bureau were elected by the Central Committee, who were in turn elected by the Soviet Congress. These bodies rarely dismissed the Premier (executive leader), but these bodies did - in theory and in some cases, in practice - worked to choose the Premier and keep said Premier in check. Both Georgy Malenkov and Khrushchev were dismissed by the Central Committee and kicked out of office. Georgy actually had his power restricted during his time as Premier - he tried to take control as Premier and secretariat of the Communist Party after Stalin's death, but nine days later, the Politburo forced him to give up his position as Secretariat and keep the position separate from premiership.
tl;dr: The Central Committee thanks to the legal framework of the USSR had regulatory power similar to the US Senate Select Committee and they could, to some degree, regulate the Premier's actions, check for abuse, and even have the Premier removed from office. It does have differences to the United States Senate Committee, such as being part of a one party state (and thus, always being part of the same political party as the Premier) and being part of a 'dictatorship of the proletariat' form of socialism, so it is not exactly a one-to-one comparsion.