Aims: We model the emergence of magnetized plasma across granular convection cells and the low atmosphere, including layers up to the mid-chromosphere. Methods: Three-dimensional numerical experiments are carried out in which the equations of MHD and radiative transfer are solved self-consistently. We use the MURaM code, which assumes local thermodynamic equilibrium between plasma and radiation. Results: In the photosphere, we find good agreement between our simulation predictions and observational results obtained with the Hinode satellite for the velocity and magnetic fields. We also confirm earlier simulation results by other authors. Our experiments reveal a natural mechanism of formation of twisted magnetic flux tubes that results from the retraction of photospheric horizontal fields at new intergranular lanes in decaying granules. In the chromosphere, we present evidence for the non-radiative heating of the emerging magnetized plasma due to the passage of shocks and/or ohmic dissipation. We study the formation of high-temperature points in the magnetic domain. We detect two types of points, classified according to whether they have a photospheric counterpart or otherwise. We also find evidence of those two types in Hinode observations. Using Lagrangian tracing of a large statistical sample of fluid particles, we detect and study episodes of convective collapse of magnetic elements returning to the photosphere. On the other hand, we study the maximum heights reached by all tracers, magnetized or otherwise. Only a small fraction (1.3%) of the magnetic elements reach the mid-chromosphere (z>750 km), while virtually no unmagnetized elements in the sample rise above the level of the reverse granulation (a few 100 km above the photosphere). We find that the rise into the chromosphere occurs in the form of successive jumps with intermediate stops rather than in a smooth continuous fashion and propose a tentative explanation of this behavior. Finally, also using Lagrange tracing, we document the creation of high-temperature points in the chromosphere via rising shock fronts.