At the beginning of 21th century, the range of solid state materials was extended by crystals featuring charge excitations with a chiral spin or pseudo-spin texture close to the Fermi energy. Such exceptional electronic properties can be found in graphene or topological insulators, which both render a great potential for upcoming electronic devices. In this thesis, mesoscopic systems of such solid state materials are investigated by a time-dependent scheme, which describes the electronic excitations by the propagation of wave packets. The time evolution of an initial state contains the information of various dynamical observables, which often feature interesting effects. For example, graphene subjected to a perpendicular magnetic field exhibits a rich revival structure, leading to a periodic sequence of collapses and revivals of an initially localized state. An additional dynamical effect arising in superstructures of zero-gap semiconductors are so called Bloch-Zener oscillations, which arise because of a Zener tunneling event between the electron and hole branch during one Bloch cycle. This leads to a richer frequency pattern and generates prominent traces in the current-voltage characteristics. Furthermore, dynamical observables can also be transformed into static physical properties, like density of states or transmission amplitudes. Accordingly, a wave-packet propagating through a mesoscopic system can be employed to derive its static scattering characteristics. This is utilized to investigate the switching functionality of constrictions based on mercury telluride heterostructures, which act as a building block for a novel charge and spin-transistor presented in this thesis. Furthermore, Berry phase effects of mercury telluride quantum wells acting as topological or conventional insulators are studied with respect to signatures in diffusive transport like weak localization.