  We consider evolution operators $G(t,s)$ associated to a class of
nonautonomous elliptic operators with unbounded coefficients, in the space of
bounded and continuous functions over $\mathbb{R}^d$. We prove some new
pointwise estimates for the spatial derivatives of the function $G(t,s)f$, when
$f$ is bounded and continuous or much smoother. We then use these estimates to
prove smoothing effects of the evolution operator in $L^p$-spaces. Finally, we
show how pointwise gradient estimates have been used in the literature to study
the asymptotic behaviour of the evolution operator and to prove summability
improving results in the $L^p$-spaces related to the so-called tight evolution
system of measures.
