  Here we generalize ideas of unified Dark Matter Dark Energy in the context of
Two Measure Theories and of Dynamical space time Theories. In Two Measure
Theories one uses metric independent volume elements and this allows to
construct unified Dark Matter Dark Energy, where the cosmological constant
appears as an integration constant associated to the equation of motion of the
measure fields. The Dynamical space time Theories generalize the Two Measure
Theories by introducing a vector field whose equation of motion guarantees the
conservation of a certain Energy Momentum tensor, which may be related, but in
general is not the same as the gravitational Energy Momentum tensor. We propose
two formulations of this idea: I - by demanding that this vector field be the
gradient of a scalar, II - by considering the dynamical space field appearing
in another part of the action. Then the Dynamical space time Theory becomes a
theory of Diffusive Unified 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 of interacting DE-DM
dust models in the form of a diffusive type interacting Unified Dark Energy and
Dark Matter scenario. We solved analytically the theories for perturbative
solution and asymptotic solution, and we show that the $\Lambda$CDM is a fixed
point of these theories at large times. Also a preliminary argument about the
good behavior of the theory at the quantum level is proposed for both theories.
