Russia did not keep all of the nuclear weapons. Ukraine and Kazakhstan inherited about 5,000 warheads and Belarus a much smaller number. Ukraine Belarus and Kazakhstan voluntarily decided to disband their nuclear programs and now no longer have any.

Of the Soviet states, Russia was the only one that had the full weapons fuel cycle and the command and control infrastructure necessary to fire and use the weapons. They also inherited the vast bulk of the Soviet stockpile. They thus were clearly the "inheritor" of the Soviet Union's nuclear capabilities and status on the whole, while the other three post-Soviet states that still possessed some weapons (Ukraine, Kazakhstan, and Belarus) only had the warheads themselves (and thus could not maintain or manufacture new weapons, and also may not have had the ability to launch them without authorization, i.e. "nuclear codes," from Moscow). To my knowledge there was no contesting of the idea that Russia was the "successor" in this respect. This was codified by the Lisbon Protocol of 1991.

The real question in the immediate post-Cold War was the status of the weapons in Ukraine, Kazakhstan, and Belarus. Again, it is not clear whether they were operational without Russian permission (there is not much out there on Russian "positive" control of their nuclear weapons from this period — it may have existed, it may not have). There were some negotiations with these states who used the "return" of these weapons as leverage for some compensation and security guarantees.

I have not seen anything that discussed the exact means by which this transfer occurred, but nuclear weapons are typically moved by guarded convoys of trucks, trains, or aircraft.