  We propose a novel leptogenesis scenario in the gauged
$U(1)_{L_{\mu}-L_{\tau}}$ model. Achieving successful leptogenesis in the
$U(1)_{L_{\mu}-L_{\tau}}$ symmetric phase is challenging due to the absence of
a CP phase, caused by restriction from the gauge symmetry. To overcome this
issue, we introduce an additional global symmetry, $U(1)_{B-L}$, and a scalar
field $\Phi$ responsible for breaking this symmetry. Through the kinetic
misalignment mechanism, the majoron field associated with $U(1)_{B-L}$ symmetry
breaking has a kinetic motion in the early universe. Subsequently,
time-dependent majoron field background induces the background CP phase
dynamically, leading to successful leptogenesis in the
$U(1)_{L_{\mu}-L_{\tau}}$ symmetric phase. Furthermore, majoron itself serves
as a dark matter candidate in this scenario. As one of the phenomenological
applications, we consider the model that can also explain the muon $g-2$
anomaly.
