  Consider a locally finite Dawson-Watanabe superprocess $\xi=(\xi_t)$ in
$\mathsf{R}^d$ with $d\geq2$. Our main results include some recursive formulas
for the moment measures of $\xi$, with connections to the uniform Brownian
tree, a Brownian snake representation of Palm measures, continuity properties
of conditional moment densities, leading by duality to strongly continuous
versions of the multivariate Palm distributions, and a local approximation of
$\xi_t$ by a stationary cluster $\tilde{\eta}$ with nice continuity and scaling
properties. This all leads up to an asymptotic description of the conditional
distribution of $\xi_t$ for a fixed $t>0$, given that $\xi_t$ charges the
$\varepsilon$-neighborhoods of some points $x_1,\ldots,x_n\in \mathsf{R}^d$. In
the limit as $\varepsilon\to0$, the restrictions to those sets are
conditionally independent and given by the pseudo-random measures $\tilde{\xi}$
or $\tilde{\eta}$, whereas the contribution to the exterior is given by the
Palm distribution of $\xi_t$ at $x_1,\ldots,x_n$. Our proofs are based on the
Cox cluster representations of the historical process and involve some delicate
estimates of moment densities.
