  We study the following model for a diploid population of constant size $N$:
Every individual carries a random number of (genetic) elements. Upon a
reproduction event each of the two parents passes each element independently
with probability $\tfrac 12$ on to the offspring. We study the process $X^N =
(X^N(1), X^N(2),...)$, where $X_t^N(k)$ is the frequency of individuals at time
$t$ that carry $k$ elements, and prove convergence (in some weak sense) of
$X^N$ jointly with its empirical first moment $Z^N$ to the ``slow-fast'' system
$(Z,X)$, where $X_t = \text{Poi}(Z_t)$ and $Z$ evolves according to a critical
Feller branching process. We discuss heuristics explaining this finding and
some extensions and limitations.
