Most of my knowledge relating to this has to do with England during the Commonwealth period, when Cromwell and his puritanical supporters overthrew the House of Stuart. In this case, the mint and treasury were seized in 1645, but they waited until they abolished the monarchy three years later to mint their own currency, producing coins identical to those in 1645. They kept the weights and values the same and replaced portraits and other images with shields on both sides.

In general, the value of coins would vary between rulers of the same line, often changing during their reign! So the real value of a coin was determined by weight. The benefit of minted coinage was that all coins of the same issue were expected to be worth the same amount. Other coins would be melted down and recast as they entered the money supply.

Source: British Museum of Coins and Metals, Handbook of the Coins of Great Britain and Ireland. 1899.

When a kingdom is overthrown by another ruler, new coins are introduced.

This varies. Not all kings issue coins. Particularly if coin use is restricted to certain areas of society (international traders say) they many not feel the need to. In other cases they may not be able to either because they lack the ability or because the population would be unlikely to accept them (you see this with the Maria Theresa Thaler in Abyssinia).

As for what happens to older coins it depends. Sometimes they will contiue to circulate alongside newer coins. Eventualy they wear out or Gresham's law kicks in and gold and silver coins end up in the melting pot for bullion value. Bronze coins suffer a simular fate although its questionable if the term bullion applies.

In other cases the new ruler will attempt to suppress coins of a previous ruler in which case if successful coins get melted down for bullion value or stored away (either in the hope that things change or so that they can later be melted down for bullion value if needed).

Another thing you see from time to time is that the new ruler introduces so many coins that the old coins become irrelevant. British Celtic coins after the Roman invasion. We have no evidence that they were formally banned but you would quickly have a hard time finding someone who would accept them so again into the melting pot they would have gone.

Then you have cases where coins pretty much drop out of use entirely. With the fall of the roman empire in britian any coin use became pretty minimal. Gold and silver coins again tend towards the melting pot (although a few were used in jewellery). Bronze would also be worth melting but a fair number may simply have been lost (the value of some of the smaller roman bronze coins wouldn't have made searching for them very worthwhile).