  We construct and study the stochastic force field generated by a Poisson
distribution of sources at finite density, $x_1,x_2,\cdots$ in $\mathbb{R}^3$
each of them yielding a long range potential $Q_i\Phi(x-x_i)$ with possibly
different charges $Q_i \in \mathbb{R}$. The potential $\Phi$ is assumed to
behave typically as $|x|^{-s}$ for large $|x|$, with $s > 1/2$. We will denote
the resulting random field as "generalized Holtsmark field". We then consider
the dynamics of one tagged particle in such random force fields, in several
scaling limits where the mean free path is much larger than the average
distance between the scatterers. We estimate the diffusive time scale and
identify conditions for the vanishing of correlations. These results are used
to obtain appropriate kinetic descriptions in terms of a linear Boltzmann or
Landau evolution equation depending on the specific choices of the interaction
potential.
