  In this paper we consider the homogenization of the evolution problem
associated with a jump process that involves three different smooth kernels
that govern the jumps to/from different parts of the domain. We assume that the
spacial domain is divided into a sequence of two subdomains $A_n \cup B_n$ and
we have three different smooth kernels, one that controls the jumps from $A_n$
to $A_n$, a second one that controls the jumps from $B_n$ to $B_n$ and the
third one that governs the interactions between $A_n$ and $B_n$.Assuming that
$\chi_{A_n} (x) \to X(x)$ weakly in $L^\infty$ (and then $\chi_{B_n} (x) \to
1-X(x)$ weakly in $L^\infty$) as $n \to \infty$ and that the initial condition
is given by a density $u_0$ in $L^2$ we show that there is an homogenized limit
system in which the three kernels and the limit function $X$ appear. When the
initial condition is a delta at one point, $\delta_{\bar{x}}$ (this corresponds
to the process that starts at $\bar{x}$) we show that there is convergence
along subsequences such that $\bar{x} \in A_{n_j}$ or $\bar{x} \in B_{n_j}$ for
every $n_j$ large enough.
  We also provide a probabilistic interpretation of this evolution equation in
terms of a stochastic process that describes the movement of a particle that
jumps in $\Omega$ according to the three different kernels and show that the
underlying process converges in distribution to a limit process associated with
the limit equation.
  We focus our analysis in Neumann type boundary conditions and briefly
describe at the end how to deal with Dirichlet boundary conditions.
