  We present a model of (double) kinetic theory which paves the way to describe
matter in a Double Field Theory background. Generalized diffeomorphisms acting
on double phase space tensors are introduced. The generalized covariant
derivative is replaced by a generalized Liouville operator as it happens in
relativistic kinetic theory. The section condition is consistently extended and
the closure of the generalized transformations is still given by the C-bracket.
In this context we propose a generalized Boltzmann equation and compute the
moments of the latter, obtaining an expression for the generalized
energy-momentum tensor and its conservation law.
