  We introduce a framework, which is a mesoscopic-fluctuation-scale analog of
Yau's method [46] for hydrodynamic limits, for deriving KPZ equations with
time-dependent coefficients from time-inhomogeneous interacting particle
systems. To our knowledge, this is the first derivation of a time-inhomogeneous
KPZ equation whose solution theory has an additional nonlinearity that is
absent in the time-homogeneous case. So, we also show global well-posedness for
the SPDE. To be concrete, we restrict to time-inhomogeneous Ginzburg-Landau
SDEs. The method for deriving KPZ is based on a Cole-Hopf transform, whose
analysis is the bulk of this paper. The key ingredient for said analysis is a
``local" second-order Boltzmann-Gibbs principle, shown by stochastic calculus
of the Ginzburg-Landau SDEs and regularity estimates for their Kolmogorov
equations, all of which likely generalizes to many other particle systems. This
addresses a ``Big Picture Question" in [47] on deriving KPZ equations. It is
also, to our knowledge, a first result on KPZ-type limits in a non-equilibrium
like that in [6].
