Anderson localization is a wave interference phenomena. The single-particle wave functions become localized for an arbitrary amount of disorder for one-dimensional disordered systems in the thermodynamic limit. The localization characteristics of disordered systems allows us to measure the sensitivity of time reversal in the presence of small perturbations, namely the Loschmidt echo, which indirectly shows dynamical phase transitions via the return rate function. For single-particle non-interacting problems, calculating the Loschmidt echo for large system sizes is straightforward. For many-particle interacting problems, however, it is computationally affordable only for small system sizes. Using both single-particle and many-particle Hamiltonian approaches, we numerically investigate the disorder-averaged Loschmidt echo and dynamical phase transitions for quantum quenches in the disordered SSH model and the disordered XXZ spin model where analytical solutions are not attainable. We show that dynamical phase transitions persist for weakly disordered systems and finite system sizes.