  We study the null-controllability of parabolic equations associated to
non-autonomous Ornstein-Uhlenbeck operators. When a Kalman type condition holds
for some positive time $T>0$, these parabolic equations are shown to enjoy a
Gevrey regularizing effect at time $T>0$. Thanks to this regularizing effect,
we prove by adapting the Lebeau-Robbiano method that these parabolic equations
are null-controllable in time greater than or equal to $T>0$ from control
regions, for which null-controllability is classically known to hold in the
case of the heat equation.
