  We extend the concept of the Kodama symmetry, a quasi-local time translation
symmetry for dynamical spherically symmetric spacetimes, to a specific class of
dynamical axisymmetric spacetimes, namely the families of Kerr-Vaidya and
Kerr-Vaidya-de Sitter spacetimes. We study some geometrical properties of the
asymptotically flat Kerr-Vaidya metric, such as the Brown-York mass and the
Einstein tensor. Furthermore, we propose a generalization of the Kerr-Vaidya
metric to an asymptotic de Sitter background. We show that for these classes of
dynamical axisymmetric black hole spacetimes, there exists a timelike vector
field that exhibits similar properties to the Kodama vector field in spherical
symmetry. This includes the construction of a covariantly conserved current and
a corresponding locally conserved charge, which in the Kerr-Vaidya case
converges to the Brown-York mass in the asymptotically flat region.
