The closest word I can think of that involves the active removal of something is recall:
1 : CANCEL, REVOKE
The equivalent to the example sentence would be:
The Royal Mint has the right to recall old coins.
However, while that verb makes the most sense to me, it's not normally used with things like coins; it's more commonly used in the context of dealing with defective merchandise.
Another possibility is decommission:
: to remove (something, such as a ship or a nuclear power plant) from service
// The government is decommissioning the nuclear power plant.
// The satellite will be decommissioned on October 10.
— David Grossman, Popular Mechanics, "Requiem for a Climate Change Satellite, Decommissioned After 11 Years," 6 Oct. 2019
The Royal Mint has the right to decommission old coins.
But that is also a word that wouldn't normally be associated with coins specifically.
While both words work, and which to choose depends on the nuance being looked for, I don't believe there's one that's commonly used in the exact situation of coins—even if both work quite well with respect to other things.
Also, neither word necessitates actual destruction. While that could certainly be the result, something that is recalled could be repaired and put back into circulation again, while something that is decommissioned could simply be removed and stored somewhere.