In principle, but also often in practice, Communist parties were connected through Comintern or Third International. In Europe it did have a lot of influence on local parties, going so far to determine key positions such as general secretaries. Idea was that as USSR was the first state of socialism, its guidance and experience would prove valuable to all other communist parties, and among many it was accepted as such. As war became more likely, they would also have to take on a role of helping to defend it. It is one of the main reasons why communists across Europe switched to Popular front policies (wide anti-fascist coalitions, not necessarily revolutionary) in the 30s.

However, there were always discussions about how relevant Soviet revolutionary experience was in a changed situation, and how closely should Comintern be followed. Where Communist Parties were illegal (much of eastern and central Europe), they were much more dependent on Moscow. I can't say much about Americas, apart that direct influence was much less, there were fractions more loyal to Comintern, but also other, most prominently Trockyite fractions.

Outside the socialist block after Second world war, Moscow influence (Comintern was disbanded in 1943 as a good will gesture towards the western allies) declined after death of Stalin and Khrushchev's "secret speech" in 1956 criticizing Stalin, personality cult and his crimes. Distancing of west European communist from Moscow was helped along by Cold war divisions, Soviet intervention in Hungary and alternative interpretations of Marxism. With China there was also another revolutionary model, considered more suitable to agricultural Third world nations, and even in some ways adopted in Europe, though more in name than any specific revolutionary strategy.

For the last point, The Wind from the East by Wolin in an excellent read, an intellectual history of 1960s left. The Red Flag by Priestland is very good intellectual, social and political history of communism.