  Using a fully covariant treatment for the description of the bulk geometry,
we study the brane cosmological evolution in the presence of a smooth bulk
matter distribution. We focus on the case of a Friedmann-Robertson-Walker (FRW)
brane, invariantly characterized by the existence of a six-dimensional group of
isometries acting on 3D spacelike orbits. With a FRW brane, the bulk geometry
can be regarded as the 5D generalization of the \emph{inhomogeneous orthogonal
family of Locally Rotationally Symmetric (LRS)} spacetimes. We show that, for
\emph{any bulk matter configuration}, the expansion rate on the brane depends
only on the covariantly defined \emph{comoving mass} $\mathcal{M}$ of the bulk
fluid within a radius equal to the average length scale of the 3D spacelike
hypersurfaces of constant curvature. This unique contribution incorporates the
effects of the 5D Weyl tensor and the projected tensor related to the bulk
matter, and gives a transparent physical picture that includes an effective
conservation equation between the brane and the bulk matter.
