  We introduce and study a multiparameter colored partition category
$\mathcal{CPar}(\textbf{x})$ by extending the construction of the partition
category, over an algebraically closed field $\Bbbk$ of characteristic zero and
for a multiparameter $\textbf{x}\in \Bbbk^{r}$. The morphism spaces in
$\mathcal{CPar}(\textbf{x})$ have bases in terms of partition diagrams whose
parts are colored by elements of the multiplicative cyclic group $C_r$. We show
that the endomorphism spaces of $\mathcal{CPar}(\textbf{x})$ and additive
Karoubi envelope of $\mathcal{CPar}(\textbf{x})$ are generically semisimple.
The category $\mathcal{CPar}(\textbf{x})$ is rigid symmetric strict monoidal
and we give a presentation of $\mathcal{CPar}(\textbf{x})$ as a monoidal
category. The path algebra of $\mathcal{CPar}(\textbf{x})$ admits a triangular
decomposition with Cartan subalgebra being equal to the direct sum of the group
algebras of complex reflection groups $G(r,n)$. We compute the structure
constants for the classes of simple modules in the split Grothendieck ring of
the category of modules over the path algebra of the downward partition
subcategory of $\mathcal{CPar}(\textbf{x})$ in two ways. Among other things,
this gives a closed formula for the product of the reduced Kronecker
coefficients in terms of the Littlewood--Richardson coefficients for $G(r,n)$
and certain Kronecker coefficients for the wreath product $(C_r \times C_r)\wr
S_n$. For $r=1$, this formula reduces to a formula for the reduced Kronecker
coefficients given by Littlewood. We also give two analogues of the
Robinson--Schensted correspondence for colored partition diagrams and, as an
application, we classify the equivalence classes of Green's left, right and
two-sided relations for the colored partition monoid in terms of these
correspondences.
