  In this paper, we discuss the chordal Komatu-Loewner equation on standard
slit domains in a manner applicable not just to a simple curve but also a
family of continuously growing hulls. Especially a conformally invariant
characterization of the Komatu-Loewner evolution is obtained. As an
application, we prove a sort of conformal invariance, or locality, of the
stochastic Komatu-Loewner evolution $\mathrm{SKLE}_{\sqrt{6},
-b_{\mathrm{BMD}}}$ in a fully general setting, which solves an open problem
posed by Chen, Fukushima and Suzuki [Stochastic Komatu-Loewner evolutions and
SLEs, Stoch. Proc. Appl. 127 (2017), 2068-2087].
