  In this paper we introduce refined canonical stable Grothendieck polynomials
and their duals with two infinite sequences of parameters. These polynomials
unify several generalizations of Grothendieck polynomials including canonical
stable Grothendieck polynomials due to Yeliussizov, refined Grothendieck
polynomials due to Chan and Pflueger, and refined dual Grothendieck polynomials
due to Galashin, Liu, and Grinberg. We give Jacobi--Trudi-type formulas,
combinatorial models, Schur expansions, Schur positivity, and dualities of
these polynomials. We also consider flagged versions of Grothendieck
polynomials and their duals with skew shapes.
