An exact time-dependent solution for the wave function $\psi(r, t)$ of a particle moving in the presence of an asymmetric rectangular well/barrier potential varying in one dimension is obtained by applying a novel for this problem approach using multiple scattering theory (MST) for the calculation of the space—time propagator. This approach, based on the found effective potentials localized at the potential jumps and responsible for transmission through and reflection from the considered rectangular potential, enables considering these processes from standpoint of a particle (rather than a wave). The solution describes these quantum phenomena as time dependent and is related to both the fundamental issues (such as time measurement) of quantum mechanics and the kinetic theory of nanostructures due to the fact that the considered potential can model the spin-dependent potential profile of the magnetic multilayers used in spintronics devices. It is presented in terms of integrals of elementary functions and is a sum of the forward- and backward-moving components of the wave packet. The relative contribution of these components and their interference as well as of the potential asymmetry to both the probability density $|\psi(r, t)|^{2}$ and the particle dwell time is considered and numerically visualized for the narrow and broad energy (momentum) distributions of the initial Gaussian wave packet. As shown, in the case of a broad initial wave packet, the quantum-mechanical counterintuitive effect of the influence of the backward-moving components on the considered quantities becomes significant (that is often disregarded). The influence of the potential asymmetry in this case can also be more pronounced.