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

Null-controllability of non-autonomous Ornstein-Uhlenbeck equations