See my preprint http://arxiv.org/abs/1410.5293 Theorem 3.2 on page 7ff.:

Let $S$ be a regular, Noetherian, integral, separated scheme, and $g: \{\eta\} \hookrightarrow S$ the inclusion of the generic point.  Let $\mathcal{A}/S$ be an Abelian scheme.  Then 
$$
    \mathcal{A} = g_*g^*\mathcal{A}
$$
as sheaves on $S_{\mathrm{sm}}$. (This is the "Néron mapping property" for $\mathcal{A}/S$.)