Critical physical issues can be specifically tackled with the global full-f gyrokinetic code GYSELA. Three main results are presented. First, the self-consistent treatment of equilibrium and fluctuations highlights the competition between two compensation mechanisms for the curvature driven vertical charge separation, namely, parallel flow and polarization. The impact of the latter on the turbulent transport is discussed. In the non-linear regime, the benchmark with the Particle-In-Cell code ORB5 looks satisfactory. Second, the transport scaling with p. is found to depend both on p. itself and on the distance to the linear threshold. Finally, a statistical steady-state turbulent regime is achieved in a reduced version of GYSELA by prescribing a constant heat source.