  We show that the evolution of isoparametric hypersurfaces of Riemannian
manifolds by the mean curvature flow is given by a reparametrization of the
parallel family in short time, as long as the uniqueness of the mean curvature
flow holds for the initial data and the corresponding ambient space. As an
application, we provide a class of Riemannian manifolds that admit
hypersurfaces with constant principal curvatures, which are not isoparametric
hypersurfaces. Furthermore, for a class of ambient spaces, we show that the
singularities developed by the mean curvature flow with isoparametric
hypersurfaces as the initial data are Type I singularities. We apply our
results to describe the evolution of isoparametric hypersurfaces by the mean
curvature flow in ambient spaces with nonconstant sectional curvature, such as
homogenous 3-manifolds $\mathbb{E}(\kappa, \tau)$ with 4-dimensional isometry
groups, and Riemannian products $\mathbb{Q}^2_{c_1} \times \mathbb{Q}^2_{c_2}$
of space forms.
