We report on the dynamics of magnetic domain structure conversions exhibited by soft magnetic thin-film elements of elementary geometrical shape (square, disc, triangle) when exposed to a strong external magnetic field. Starting from flux closure vortex patterns, the magnetic structures evolve towards an in-plane saturated state under the influence of an external field. This irreversible and nucleation-free magnetization process occurs on the time scale of picoseconds. The details of this conversion are investigated by means of a time-resolved micromagnetic finite element modeling. We find a sensitive dependence of the temporal evolution of the magnetic structure on the value of the damping parameter in Gilbert's equation of motion. In the case of high damping, domain wall motion dominates the process, while lower damping leads to the formation of a 360° wall which collapses by emitting magnetization waves. It is shown that the mobility of vortices is generally much lower than that of domain walls. The calculations indicate that at a low damping, a magnetic vortex can act almost as a source for concentric waves in ferromagnetic thin-film elements.