  We develop the kinetic theory of collisionless relaxation for systems with
long-range interactions in relation to the statistical theory of Lynden-Bell.
We treat the multi-level case. We make the connection between the kinetic
equation obtained from the quasilinear theory of the Vlasov equation and the
relaxation equation obtained from a maximum entropy production principle. We
propose a method to close the infinite hierarchy of kinetic equations for the
phase level moments and obtain a kinetic equation for the coarse-grained
distribution function in the form of a generalized Landau, Lenard-Balescu or
Kramers equation associated with a generalized form of entropy [P.H. Chavanis,
Physica A {\bf 332}, 89 (2004)]. This allows us to go beyond the two-level case
associated with a Fermi-Dirac-type entropy. We discuss the numerous analogies
with two-dimensional turbulence. We also mention possible applications of the
present formalism to fermionic and bosonic dark matter halos.
