The Yellowstone magmatic system is one of the largest magmatic systems on Earth, and thus an ideal location to study magmatic processes. Whereas previous seismic tomography results could only image a shallow magma chamber, a recent study using more seismometers showed that a second and massive partially molten mush chamber exists above the Moho (Huang et al., 2015). To understand the mechanics of this system, it is thus important to take the whole system from the mantle plume up to the shallow magma chambers into account. Here, we employ lithospheric-scale 3D visco-elasto-plastic geodynamic models to test the influence of parameters such as the connectivity of the chambers and rheology of the lithosphere on the dynamics of the system. A gravity inversion is used to constrain the effective density of the magma chambers, and an adjoint modelling approach reveals the key model parameters affecting the surface velocity. Model results show that a combination of connected chambers with plastic rheology can explain the recorded slow vertical surface uplift rates of around 1.2 cm/a, as representing a long term background signal. A geodynamic inversion to fit the model to observed GPS surface velocities, reveals that the magnitude of surface uplift varies strongly with the viscosity difference between the chambers and the crust. Even though stress directions have not been used as inversion parameter, modelled stress orientations are consistent with observations. However, phases of larger uplift velocities can also result from magma inflation which is a short term effect. We consider two approaches: 1) overpressure in the magma chamber in the asthenosphere and 2) inflation of the uppermost chamber prescribed by an internal kinematic boundary condition. We demonstrate that the asthenosphere inflation has a smaller effect on the surface velocoties in comparison with the uppermost chamber inflation. We show that the pure buoyant uplift of magma bodies in combination with magma inflation can explain (varying) observed uplift rates at the example of the Yellowstone volcanic system.