  Generalized ideas of unified dark matter and dark energy in the context of
dynamical space time theories with a diffusive transfer of energy are studied.
The dynamical space-time theories are introduced a vector field whose equation
of motion guarantees a conservation of a certain Energy Momentum tensor, which
may be related, but in general is not the same as the gravitational Energy
Momentum tensor. This particular energy momentum tensor is built from a general
combination of scalar fields derivatives as the kinetic terms, and possibly
potentials for the scalar field. By demanding that the dynamical space vector
field be the gradient of a scalar the dynamical space time theory becomes a
theory for diffusive interacting dark energy and dark matter. These
generalizations produce non-conserved energy momentum tensors instead of
conserved energy momentum tensors which leads at the end to a formulation for
interacting DE-DM. We solved analytically the theories and we show that the
$\Lambda$CDM is a fixed point of these theories at large times. A particular
case has asymptotic correspondence to previously studied non-Lagrangian
formulations of diffusive exchange between dark energy dark matter.
