  Motion groups of links in the three sphere $\mathbb{S}^3$ are generalizations
of the braid groups, which are motion groups of points in the disk
$\mathbb{D}^2$. Representations of motion groups can be used to model
statistics of extended objects such as closed strings in physics. Each
$1$-extended $(3+1)$-topological quantum field theory (TQFT) will provide
representations of motion groups, but it is difficult to compute such
representations explicitly in general. In this paper, we compute
representations of the motion groups of links in $\mathbb{S}^3$ with
generalized axes from Dijkgraaf-Witten (DW) TQFTs inspired by dimension
reduction. A succinct way to state our result is as a step toward a twisted
generalization (Conjecture \ref{mainconjecture}) of a conjecture for DW
theories of dimension reduction from $(3+1)$ to $(2+1)$: $\textrm{DW}^{3+1}_G
\cong \oplus_{[g]\in [G]} \textrm{DW}^{2+1}_{C(g)}$, where the sum runs over
all conjugacy classes $[g]\in [G]$ of $G$ and $C(g)$ the centralizer of any
element $g\in [g]$. We prove a version of Conjecture \ref{mainconjecture} for
the mapping class groups of closed manifolds and the case of torus links
labeled by pure fluxes.
