  The nested Kingman coalescent describes the dynamics of particles (called
genes) contained in larger components (called species), where pairs of species
coalesce at constant rate and pairs of genes coalesce at constant rate provided
they lie within the same species. We prove that starting from $rn$ species, the
empirical distribution of species masses (numbers of genes$/n$) at time $t/n$
converges as $n\to\infty$ to a solution of the deterministic
coagulation-transport equation $$
  \partial_t d \ = \ \partial_x ( \psi d ) \ + \ a(t)\left(d\star d - d
\right), $$ where $\psi(x) = cx^2$, $\star$ denotes convolution and $a(t)=
1/(t+\delta)$ with $\delta=2/r$. The most interesting case when $\delta =0$
corresponds to an infinite initial number of species. This equation describes
the evolution of the distribution of species of mass $x$, where pairs of
species can coalesce and each species' mass evolves like $\dot x = -\psi(x)$.
We provide two natural probabilistic solutions of the latter IPDE and address
in detail the case when $\delta=0$. The first solution is expressed in terms of
a branching particle system where particles carry masses behaving as
independent continuous-state branching processes. The second one is the law of
the solution to the following McKean-Vlasov equation $$ dx_t \ = \ - \psi(x_t)
\,dt \ + \ v_t\,\Delta J_t $$ where $J$ is an inhomogeneous Poisson process
with rate $1/(t+\delta)$ and $(v_t; t\geq0)$ is a sequence of independent rvs
such that $ {\mathcal L}(v_t) = {\mathcal L}(x_t)$. We show that there is a
unique solution to this equation and we construct this solution with the help
of a marked Brownian coalescent point process. When $\psi(x)=x^\gamma$, we show
the existence of a self-similar solution for the PDE which relates when
$\gamma=2$ to the speed of coming down from infinity of the nested Kingman
coalescent.
