As you may know, most Central Banks in the world are not owned by the state/government. Rather they are privately owned by bank cartels. Or independent, or whatever euphemism they use to mask the true state of the Central Bank.
A Central Bank which is owned by the state is called a nationalized bank. For example, in post colonial era, in the midst of the 20th century, many of the new born nations who gained independence would nationalize their Central Bank from their previous colonial rulers ownership.
The bank is then owned by the state and answers to the constitution or government body. While an INDEPENDENT or privately owned bank would answer to their rightfully private owners. Which most likely are a chain of, in turn also, privately owned corporations in the form of banks as well.
My question is, how did those banks gain ownership of a Central Bank? According what law was it possible to acquire possession of a Central Bank? What and how was the mechanism? Which set of law was used? Whom did they pay the tag price to?
I don't think you understand how a central banking system works, because very few if any work in the way you describe. Central banking systems are created typically through government action as a means of increasing financial liquidity (or the ability to lend and borrow money) as well as acting as a lender of last resort in times of crisis to help stave off full economic collapse. Despite this, they are independent of the government as otherwise it is possible for the government to pressure the bank into making monetary policies that would benefit the government's fiscal policy, for instance in the case of Nazi Germany, the bank helped propagate Mefos bills which were used to help Germany rearm at the expense of her long-term economic viability (leading to potential German economic collapse in April 1939).