We study how well magnetic models can be implemented with ultracold bosonic atoms of two different hyperfine states in an optical superlattice. The system is captured by a two-species Bose-Hubbard model, but realizes in a certain parameter regime actually the physics of a spin-1/2 Heisenberg magnet, describing the second order hopping processes. Tuning of the superlattice allows for controlling the effect of fast first order processes versus the slower second order ones. Using the density-matrix renormalization-group method, we provide the evolution of typical experimentally available observables. The validity of the description via the Heisenberg model, depending on the parameters of the Hubbard model, is studied numerically and analytically. The analysis is also motivated by recent experiments [S. Foelling et al., Nature 448, 1029 (2007); S. Trotzky et al., Sience 319, 295 (2008)] where coherent two-particle dynamics with ultracold bosonic atoms in isolated double wells were realized. We provide theoretical background for the next step, the observation of coherent many-particle dynamics after coupling the double wells. Contrary to the case of isolated double wells, relaxation of local observables can be observed. The tunability between the Bose-Hubbard model and the Heisenberg model in this setup could be used to study experimentally the differences in equilibration processes for nonintegrable and Bethe ansatz integrable models. We show that the relaxation in the Heisenberg model is connected to a phase averaging effect, which is in contrast to the typical scattering driven thermalization in nonintegrable models. We discuss the preparation of magnetic groundstates by adiabatic tuning of the superlattice parameters.