The presence of a large external magnetic field in a plasma introduces an additional time-scale which is very constraining for the numerical simulation. Hence it is very useful to introduce averaged models which remove this time-scale. However, depending on other parameters of the plasma different starting models for the asymptotic analysis may be considered- . We introduce here a generic framework for our analysis which fits many of the possible regimes and apply it in particular to justify the Finite Larmor Radius approximation both in the linear case and in the non linear case in the plane transverse to the magnetic field.