  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 $\mathcal{B}$.
Our results significantly extend the pathwise approximation for
non-regime-switching, time-homogeneous SDE in R\"omisch et al. (1985). Here,
two key techniques are the Lamperti transform for regime-switching SDEs and a
pathwise analysis of their solutions.
