A characteristic feature of fluid theories concerns the difficulty of uniquely defining consistent closure conditions for the fluid equations. In fact it is well known that fluid theories cannot generally provide a closed system of equations for the fluid fields. This feature is typical of collisionless plasmas where, in contrast to collisional plasmas, asymptotic closure conditions do not follow as a consequence of an H-theorem This issue is of particular relevance in astrophysics where fluid approaches are usually adopted. On the other hand, it is well known that the determination of the closure conditions is in principle achievable in the context of kinetic theory. In the case of multi-species thermal magnetoplasmas this requires the determination of the species tensor pressure and of the corresponding heat fluxes. In this paper we investigate this problem in the framework of the Vlasov-Maxwell description for collisionless axisymmetric magnetoplasmas arising in astrophysics, with particular reference to accretion discs around compact objects (like black holes and neutron stars). The dynamics of collisionless plasmas in these environments is determined by the simultaneous presence of gravitational and magnetic fields, where the latter may be both externally produced and self-generated by the plasma currents. Our starting point here is the construction of a solution for the stationary distribution function describing slowly-varying gyrokinetic equilibria. The treatment is applicable to non-relativistic axisymmetric systems characterized by temperature anisotropy and differential rotation flows. It is shown that the kinetic formalism allows one to solve the closure problem and to consistently compute the relevant fluid fields with the inclusion of finite Larmor-radius effects. The main features of the theory and relevant applications are discussed.