  We show that a multiple commutation relation of the Yang-Baxter algebra of
integrable lattice models derived by Shigechi and Uchiyama can be used to
connect two types of Grothendieck classes by the $K$-theoretic pushforward from
the Grothendieck group of Grassmann bundles to the Grothendieck group of a
nonsingular variety. Using the commutation relation, we show that two types of
partition functions of an integrable five-vertex model, which can be explicitly
described using skew Grothendieck polynomials, and can be viewed as
Grothendieck classes, are directly connected by the $K$-theoretic pushforward.
We show that special cases of the pushforward formula which correspond to the
nonskew version are also special cases of the formulas derived by Buch. We also
present a skew generalization of an identity for the Grothendieck polynomials
by Guo and Sun, which is an extension of the one for Schur polynomials by
Feh\'er, N\'emethi and Rim\'anyi. We also show an application of the
pushforward formula and derive an integration formula for the Grothendieck
polynomials.
