  We consider dynamical graphs, namely graphs that evolve over time, and
investigate a notion of local weak convergence that extends naturally the usual
Benjamini-Schramm local weak convergence for static graphs. One of the
well-known results of Benjamini-Schramm local weak convergence is that of the
inhomogeneous random graph $IRG_n(\kappa)$ on $n$ vertices with connection
kernel $\kappa$. When the kernel satisfies the mild technical condition of
being a graphical kernel, the $IRG_n(\kappa)$ converges locally in probability
to the (unimodular) multi-type (marked) Poisson-Galton-Watson tree
$MPGW(\kappa)$. We extend this to dynamical settings, by introducing the
dynamical inhomogeneous random graph $DIRG_n(\kappa,\beta),$ with connection
kernel $\kappa$ and updating kernel $\beta$, and its limit the
growth-and-segmentation multi-type Poisson-Galton-Watson tree
$GSMPGW(\kappa,\beta)$. We obtain similarly the local limit of a slightly
different dynamical model, namely the vertex updating inhomogeneous random
graph. Our framework provides a natural tool for the study of processes defined
on these graphs, that evolve simultaneously as the graph itself and with local
dynamics. We discuss briefly the case of the contact process, where we obtain a
slight reinforcement of the results of Jacob, Linker and M\"{o}rters (see
arXiv:1807.09863 and arXiv:2206.01073).
