A model for the $bar K d to pi Y N$ reactions with $Y=Lambda, Sigma$ is developed, aiming at establishing the low-lying $Lambda$ and $Sigma$ hyperon resonances through analyzing the forthcoming data from the J-PARC E31 experiment. The off-shell amplitudes generated from the dynamical coupled-channels (DCC) model, which was developed in Kamano et al. [Phys. Rev. C 90, 065204 (2014)], are used as input to the calculations of the elementary $bar K N to bar K N$ and $bar K N to pi Y$ subprocesses in the $bar K d to pi Y N$ reactions. It is shown that the cross sections for the J-PARC E31 experiment with a rather high incoming-$bar{K}$ momentum, $|vec p_{bar K}|= 1$ GeV, can be predicted reliably only when the input $bar K N to bar K N$ amplitudes are generated from a $bar KN$ model, such as the DCC model used in this investigation, which describes the data of the $bar K N$ reactions at energies far beyond the $bar K N$ threshold. We find that the data of the threefold differential cross section $dsigma/(dM_{piSigma}dOmega_{p_n})$ for the $K^- d to pi Sigma n$ reaction below the $bar K N$ threshold can be used to test the predictions of the resonance poles associated with $Lambda(1405)$. We also find that the momentum dependence of the threefold differential cross sections for the $K^- d to pi^- Lambda p$ reaction can be used to examine the existence of a low-lying $J^P=1/2^+$ $Sigma$ resonance with a pole mass $M_R = 1457 -i39$ MeV, which was found from analyzing the $K^-p$ reaction data within the employed DCC model.
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