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We consider the Higgsstrahlung process in hadronic collisions and present thecomputation of next-to-next-to-leading order predictions matched to partonshowers for both production and $H\to b\bar{b}$ decay employing theMiNNLO$_{\rm PS}$ method. We present predictions for $ZH$ and $W^{\pm}H$production including spin correlations and off-shell effects by calculating thefull processes $pp \to \ell^+\ell^-H \to \ell^+\ell^-b\bar{b}$, $pp \to\nu_\ell\bar\nu_\ell H \to \nu_\ell\bar\nu_\ell b\bar{b}$ and $pp \to \ell^\pm\nu_\ell H \to \ell^\pm \nu_\ell b\bar{b}$ in the narrow-width approximationfor the Higgs boson. For the $W^{\pm}H$ process, NNLO+PS accuracy in productionand decay is achieved for the first time. Our calculations are validatedagainst earlier simulations in the NNLOPS approach that includes NNLOcorrections via multi-differential reweighting. The new MiNNLO$_{\rm PS}$generators for these processes, which evaluate NNLO corrections on-the-fly inthe event generation, will supersede those earlier calculations. Ourpredictions are in good agreement with recent measurements of theHiggsstrahlung cross sections.

Next-to-next-to-leading order event generation for $VH$ production with $H\to b\bar{b}$ decay