If the question specifically asks about how Moscow became more influential than Kiev, the answer is in the history of Eastern politics. First of all, Kiev never was a "Russian" city per say. Yes, it was the center of eastern Slavs and the center of Slavic civilization for thousands of years. Yes, it introduced Christianity to that part of the world and was considered to be the first state in the Slavic world. However, it never was "Russian" in that "Russians" as we know them now didn't emerge as an entity until much later in time.
Moscow as a state was established by a small clan of northern Slavs led by Rurikoviches. In time it became relatively strong and Moscow began to compete against surrounding city states such as Kiev, Novgorod and others. As a result, Kiev and Moscow were always competing parties. The peak of this competition came during the reign of the most famous of Rurikoviches, Ivan the Terrible, who was able to conquer most of the Slavic cities and some Lithuanian, with the exception of Kiev. Kiev was always coveted by the Russian Empire, mostly for the greater agricultural and labor resources that were surrounding the city. Later the passage to the Black Sea became crucial since Russians were land locked.
The dream of Russian kings and queens to add Ukraine to the collection of their trophies didn't come through until 1648, when Ukrainian Ccossacks, being worn out by constant Polish invaders, had to make the choice between Catholic Pols and Orthodox Russians. The question of religion prevailed and they chose to join Russia. Since then, Russia did anything possible to diminish the importance of Kiev and Ukraine. They even came up with a different name for Ukrainians, calling them "small Russians". Ukrainians on the other hand never stopped fighting the Russians.
During the time of the Bolshevik revolution of 1917, Ukrainian people revolted against the Russian Empire and were able to establish an independent Ukrainian state. Unfortunately the communist invasion of 1919 ended Ukraine's independence and unleashed a streak of bloodshed and genocide against Ukrainians. Kiev could never become a capitol of the Soviet Union for the simple reason that doing so would mean having to acknowledge the importance of Ukraine. Neither Lenin nor Stalin would do that, as they didn't consider Ukrainians to be human beings.
Lenin bargained at the end of WWI by giving away the Ukrainian land to Germans. Stalin caused the two largest famines in the history of the world on the territory of Ukraine and starved to death approximately 3 million people. Because of this, I think the answer to why Moscow was so dominant is obvious: it builds its power on the bones of others. This is past history of course, but I don't think that many things changed since then.