  We construct Wong--Zakai approximations of time--inhomogeneous stochastic
differential equations with regime switching (RSSDEs), and provide a
convergence rate. %Given a family of finite-variation processes
$\{\mathcal{F}^{\lambda}\}_{\lambda\ge 0}$ that converge strongly to a standard
Brownian motion $\mathcal{B}$, we construct pathwise approximations for
regime-switching, time-inhomogeneous stochastic differential equations in the
Wong-Zakai sense. Moreover, we determine the rate of strong convergence to the
solutions of such regime-switching SDEs, showing that this rate is almost as
good as that of $\{\mathcal{F}^{\lambda}\}_{\lambda\ge 0}$ to $\mathcal{B}$. In
the proposed approximations, the standard Brownian motion driving the
time-inhomogeneous RSSDEs is replaced by a family of finite--variation
processes $\{\mathcal{F}^{\lambda}\}_{\lambda > 0}$. We show that if
$\mathcal{F}^{\lambda}$ strongly converges to $\mathcal{B}$ at rate
$\delta(\lambda)$, then the Wong--Zakai approximation strongly converges to the
original solution of the time--inhomogeneous RSSDE at rate $\delta(\lambda)
\lambda^{\varepsilon}$, for any $\varepsilon > 0$. This is the first paper on
Wong--Zakai approximations for time--inhomogeneous RSSDEs, and significantly
extends the counterparts for time--homogeneous SDEs without regime switching in
R\"{o}misch and Wakolbinger (1985).
