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The availability of collisional rate coefficients is a prerequisite for an accurate interpretation of astrophysical observations, since the observed media often harbour densities where molecules are populated under non--LTE conditions. In the current study, we present calculations of rate coefficients suitable to describe the various spin isomers of multiply deuterated ammonia, namely the ND$_2$H and ND$_3$ isotopologues. These calculations are based on the most accurate NH$_3$--H$_2$ potential energy surface available, which has been modified to describe the geometrical changes induced by the nuclear substitutions. The dynamical calculations are performed within the close--coupling formalism and are carried out in order to provide rate coefficients up to a temperature of $T$ = 50K. For the various isotopologues/symmetries, we provide rate coefficients for the energy levels below $\sim$ 100 cm$^{-1}$. Subsequently, these new rate coefficients are used in astrophysical models aimed at reproducing the NH$_2$D, ND$_2$H and ND$_3$ observations previously reported towards the prestellar cores B1b and 16293E. We thus update the estimates of the corresponding column densities and find a reasonable agreement with the previous models. In particular, the ortho--to--para ratios of NH$_2$D and NHD$_2$ are found to be consistent with the statistical ratios.