Context. The thermal evolution of neutron stars in soft X-ray transients (SXTs) is sensitive to the equation of state, nucleon superfluidity, and the composition and structure of the crust. Carrying out comparisons of the observations of their crust cooling with simulations offers a powerful tool for verifying theoretical models of dense matter.Aims. We study the effect of physics input on the thermal evolution of neutron stars in SXTs. In particular, we consider different modern models of the sources of deep crustal heating during accretion episodes and the effects brought on by impurities embedded in the crust during its formation.Methods. We simulated the thermal structure and evolution of episodically accreting neutron stars under different assumptions regarding the crust composition and on the distribution of heat sources and impurities. For the non-accreted crust, we considered the nuclear charge fluctuations that arise at crust formation. For the accreted crust, we compared different theoretical models of composition and internal heating. We also compared the results of numerical simulations to observations of the crust cooling in SXT MXB 1659−29.Results. The non-accreted part of the inner crust of a neutron star can have a layered structure, with almost pure crystalline layers interchanged with layers composed of mixtures of different nuclei. The latter layers have relatively low thermal conductivities, which has an effect on the thermal evolution of the transients. The impurity distribution in the crust strongly depends on models of the dense matter and the crust formation scenario. The shallow heating that is needed to reach an agreement between the theory and the observations depends on characteristics of the crust and envelope.