How much influence does Russia have over its former satellites? What is the source of that influence?

The source of influence still comes form the supply of energy.

In terms of the CIS countries, Russia has managed to destroy its relationship with Ukraine, Georgia, Moldova, Belorussia and the Baltic states. Russian soft influence has decreased dramatically in these areas. It has also pushed these countries further towards EU.Even Kazkstan is developing closer ties with US and China and Azerbaijan has declined to purchase Russian gas.