A siege is by nature a battle plan that takes months to see to its end. So it may well have been begun long before the destruction of the ring, but ended with it. As to why the Easterlings were not able to extricate the Dwarves before the ring was cast into the flames, I think that has to do with both the nature of siege and Dwarves, the heartiest and stubbornest of the Children of Eru.
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