We present results for transport properties (diffusion and viscosity) using computer simulations. Focus is made on a densified binary sodium disilicate 2SiO2–Na2O (NS2) liquid and on multicomponent magmatic liquids (MORB, basalt). In the NS2 liquid, results show that a certain number of anomalies appear when the system is densified: a diffusivity maxima/minima is found for the network-forming ions (Si,O) which bears some striking similarities with the well-known case of water or liquid silica, whereas the sodium atom displays three distinct régimes for diffusion. Some of these features can be correlated with the obtained viscosity anomaly under pressure, the latter being fairly well reproduced from the simulated diffusion constant. In model magmas (MORB liquid), we find a plateau followed by a continuous increase of the viscosity with pressure. Finally, having computed both diffusion and viscosity independently, we can discuss the validity of the Eyring equation for viscosity which relates diffusion and viscosity. It is shown that it can be considered as valid in melts with a high viscosity. On the overall, these results highlight the difficulty of establishing a firm relationship between dynamics, structure and thermodynamics in complex liquids.