  We introduce a new formulation for differential equation describing dynamics
of measures on an Euclidean space, that we call Measure Differential Equations
with sources. They mix two different phenomena: on one side, a transport-type
term, in which a vector field is replaced by a Probability Vector Field, that
is a probability distribution on the tangent bundle; on the other side, a
source term. Such new formulation allows to write in a unified way both
classical transport and diffusion with finite speed, together with creation of
mass.
  The main result of this article shows that, by introducing a suitable
Wasserstein-like functional, one can ensure existence of solutions to Measure
Differential Equations with sources under Lipschitz conditions. We also prove a
uniqueness result under the following additional hypothesis: the measure
dynamics needs to be compatible with dynamics of measures that are sums of
Dirac masses.
