My understanding is that euro banknotes circulate in a similar fashion within the euro area and that any national central bank can re-issue fit euro banknotes regardless of the original issuer.
Although the country that originated the production order is indicated by the letter prefix to the euro banknotes’ serial number, even that designation is becoming less important as the Eurosystem moves to a pooled system of printing banknotes. Even though the Eurosystem contains the central banks of sovereign nations, the unified treatment of euro banknotes recognizes the unifying force of a single currency.