  Grothendieck polynomials, introduced by Lascoux and Sch\"utzenberger, are
certain $K$-theory representatives for Schubert varieties. Symplectic
Grothendieck polynomials, described more recently by Wyser and Yong, represent
the $K$-theory classes of orbit closures for the complex symplectic group
acting on the complete flag variety. We prove a transition formula for
symplectic Grothendieck polynomials and study their stable limits. We show that
each of the $K$-theoretic Schur $P$-functions of Ikeda and Naruse arises from a
limiting procedure applied to symplectic Grothendieck polynomials representing
certain "Grassmannian" orbit closures.
