  In the context of the Standard Model effective field theory (SMEFT) the
next-to-next-to-leading (NNLO) QCD corrections to the Higgsstrahlungs ($Vh$)
processes in hadronic collisions are calculated and matched to a parton shower
(PS). NNLO+PS precision is achieved for the complete sets of SMEFT operators
that describe the interactions between the Higgs and two vector bosons and the
couplings of the Higgs, a $W$ or a $Z$ boson, and light fermions. A POWHEG-BOX
implementation of the computed NNLO SMEFT corrections is provided that allows
for a realistic exclusive description of $Vh$ production at the level of
hadronic events. This feature makes it an essential tool for future Higgs
characterisation studies by the ATLAS and CMS collaborations. Utilising our new
Monte Carlo code the numerical impact of NNLO$+$PS corrections on the kinematic
distributions in $pp \to Zh \to \ell^+ \ell^- h$ production is explored,
employing well-motivated SMEFT benchmark scenarios.
