  Given a Riemannian manifold $M,$ and an open interval $I\subset\mathbb{R},$
we characterize nontrivial totally umbilical hypersurfaces of the product
$M\times I$ -- as well as of warped products $I\times_\omega M$ -- as those
which are local graphs built on isoparametric families of totally umbilical
hypersurfaces of $M.$ By means of this characterization, we fully extend to
$\mathbb{S}^n\times\mathbb{R}$ and $\mathbb{H}^n\times\mathbb{R}$ the results
by Souam and Toubiana on the classification of totally umbilical hypersurfaces
of $\mathbb{S}^2\times\mathbb{R}$ and $\mathbb{H}^2\times\mathbb{R}.$ It is
also shown that an analogous classification holds for arbitrary warped products
$I\times_\omega\mathbb{S}^n$ and $I\times_\omega\mathbb{H}^n.$
