  In this article, we establish embeddings {\`a} la Lions and transfer of
regularity {\`a} la Bouchut for a large scale of kinetic spaces. We use them to
identify a notion of weak solutions to Kolmogorov-Fokker-Planck equations with
(local or integral) diffusion and rough (measurable) coefficients under minimal
requirements. We prove their existence and uniqueness for a large class of
source terms, first in full space for the time, position and velocity variables
and then for the kinetic Cauchy problem on infinite and finite time intervals.
