  We prove a type-uniform Chevalley formula for multiplication with divisor
classes in the equivariant quantum $K$-theory ring of any cominuscule flag
variety $G/P$. We also prove that multiplication with divisor classes
determines the equivariant quantum $K$-theory of arbitrary flag varieties.
These results prove a conjecture of Gorbounov and Korff concerning the
equivariant quantum $K$-theory of Grassmannians of Lie type A.
