Atomistic computer simulations based on the use of the conjugate gradient and molecular dynamics methods were employed to determine the core structure and dynamics of the a/2[100] edge and screw dislocations in Fe-Ni-Cr and Fe-Ni-Cr-N austenites. The embedded-atom method was used to quantify the interactions between iron, nickel, chromium and nitrogen atoms. In Fe-Ni-Cr austenite, both the edge and screw dislocations dissociate along one of the {111} planes, forming stacking fault ribbons. The ribbon widths were found to be comparable to their values calculated using the continuum theory. The analysis of dislocation dynamics showed that the phonon drag interferes more with the motion of screw dislocations, reducing their mobility in comparison with the mobility of edge dislocations. In Fe-Ni-Cr-N austenite, the structure of the dislocation core of the a/2[110] edge dislocation does not seem to be significantly affected by the presence of nitrogen. In sharp contrast, the core structure of the dissociated a/2[110] screw dislocation undergoes a major change, resulting in spreading of the core on to two or more non-parallel planes. As a result, mobility of the screw dislocations is substantially lower than that of the edge dislocations. This finding is consistent with the experimental observations that the dislocations are predominantly of the screw character in Fe-Ni-Cr-N austenite.