  We prove multi-point correlation bounds in $\mathbb{Z}^d$ for arbitrary
$d\geq 1$ with symmetrized distances, answering open questions proposed by
Sims-Warzel \cite{SW} and Aza-Bru-Siqueira Pedra \cite{ABP}. As applications,
we prove multi-point correlation bounds for the Ising model on $\mathbb{Z}^d$,
and multi-point dynamical localization in expectation for uniformly localized
disordered systems, which provides the first examples of this conjectured
phenomenon by Bravyi-K\"onig \cite{BK}.
