Kirillov and Naruse have constructed double Grothendieck polynomials to represent the equivariant K-theory classes of Schubert varieties in the complete flag manifolds of types B, C, and D. We derive a recursive formula for these polynomials, extending certain K-theoretic transition equations known in type A to all classical types. As an application, we obtain an identity that expands the K-Stanley symmetric functions in types B, C, and D into positive linear combinations of K-theoretic Schur P- and Q-functions. We also resolve several positivity conjectures related to the skew generalizations of the latter functions.