Historically, how (if at all) did House Lannister control the flow of gold and silver from their vassal’s mines/vaults into the Westerosi and wider Essosi markets? We know that controlling the availability of these precious metals is of paramount importance, yet given the environment the story plays out in, could this even be done? Have these methods changed since the unification of the Seven Kingdoms under House Targaryen? – Thank You, RSAfan.

Good question!

Well, to a large extent, the Lannisters can pull a De Beers: since they have the largest supply, they can set the price by restricting or loosening the flow from their own vaults. 

Now, in IRL, gold mines required royal licenses. Given that different Houses have their own mines in the Westerlands, it doesn’t seem to have been the case that House Lannister had that kind of system. (Maybe the De Beers technique was good enough that they didn’t have to?)