  In this paper, we find the fusion rules for the free wreath product quantum
groups $\mathbb{G}\wr_*S_N^+$ for all compact matrix quantum groups of Kac type
$\mathbb{G}$ and $N\ge4$. This is based on a combinatorial description of the
intertwiner spaces between certain generating representations of
$\mathbb{G}\wr_*S_N^+$. The combinatorial properties of the intertwiner spaces
in $\mathbb{G}\wr_*S_N^+$ then allows us to obtain several probabilistic
applications. We then prove the monoidal equivalence between
$\mathbb{G}\wr_*S_N^+$ and a compact quantum group whose dual is a discrete
quantum subgroup of the free product $\widehat{\mathbb{G}}*\widehat{SU_q(2)}$,
for some $0<q\le1$. We obtain as a corollary certain stability results for the
operator algebras associated with the free wreath products of quantum groups
such as Haagerup property, weak amenability and exactness.
