  An evolving Riemannian manifold $(M,g_t)_{t\in I}$ consists of a smooth
$d$-dimensional manifold $M$, equipped with a geometric flow $g_t$ of complete
Riemannian metrics, parametrized by $I=(-\infty,T)$. Given an additional
$C^{1,1}$ family of vector fields $(Z_t)_{t\in I}$ on $M$. We study the family
of operators $L_t=\Delta_t +Z_t $ where $\Delta_t$ denotes the Laplacian with
respect to the metric $g_t$. We first give sufficient conditions, in terms of
space-time Lyapunov functions, for non-explosion of the diffusion generated by
$L_t$, and for existence of evolution systems of probability measures
associated to it. Coupling methods are used to establish uniqueness of the
evolution systems under suitable curvature conditions. Adopting such a unique
system of probability measures as reference measures, we characterize
supercontractivity, hypercontractivity and ultraboundedness of the
corresponding time-inhomogeneous semigroup. To this end, gradient estimates and
a family of (super-)logarithmic Sobolev inequalities are established.
