Unlike conventional first-order phase transitions, the kinetics of amorphous-amorphous transitions has been much less studied. The ultrasonic experiments on the transformations between low-density and high-density amorphous ice induced by pressure or heating provided the pressure and temperature dependencies of elastic moduli. In this article, we make an attempt to build a microscopic picture of these experimentally studied transformations using the molecular dynamics method with the TIP4P/Ice water model. We study carefully the dependence of the results of elastic constants calculations on the shear and compression rates. The system size effects are considered as well. The comparison with the experimental data enriches our understanding of the transitions observed. Our modelling gives new information on the nucleation mechanisms during transformations between low-density and high-density amorphous ices and, thus, helps answering the question about first-order nature of this transition.