  We construct bundles $E_k(\A,\F) \to M$ over the complement $M$ of a complex
hyperplane arrangement \A, depending on an integer $k \geq 1$ and a set
$\F=\{f_1, \ldots, f_\mu\}$ of continuous functions $f_i \colon M \to \C$ whose
differences are nonzero on $M$, generalizing the configuration space bundles
arising in the Lawrence-Krammer-Bigelow representation of the pure braid group.
We display such families \F\ for rank two arrangements, reflection arrangements
of types $A_\ell$, $B_\ell$, $D_\ell$, $F_4$, and for arrangements supporting
multinet structures with three classes, with the resulting bundles having
nontrivial monodromy around each hyperplane. The construction extends to
arbitrary arrangements by pulling back these bundles along products of
inclusions arising from subarrangements of these types.
  We then consider the faithfulness of the resulting representations of the
arrangement group $\pi_1(M)$. We describe the kernel of the product $\rho_\X
\colon G \to \prod_{S \in \X} G_S$ of homomorphisms of a finitely-generated
group $G$ onto quotient groups $G_S$ determined by a family \X\ of subsets of a
fixed set of generators of $G$, extending a result of T.~Stanford about
Brunnian braids. When the projections $G \to G_S$ split in a compatible way, we
show the image of $\rho_\X$ is normal with free abelian quotient, and identify
the cohomological finiteness type of $G$. These results apply to some
well-studied arrangements, implying several qualitative and residual properties
of $\pi_1(M)$, including an alternate proof of a result of Artal, Cogolludo,
and Matei on arrangement groups and Bestvina-Brady groups, and a dichotomy for
a decomposable arrangement \A: either $\pi_1(M)$ has a conjugation-free
presentation or it is not residually nilpotent.
