  We expand the study of generalized brane cosmologies by allowing for an
$f(\tilde{\cal R})$ gravity term on the brane, with $\tilde{\cal R}$ the
curvature scalar derived from the induced metric. We also include arbitrary
matter components on the brane and in the five-dimensional bulk. At low
energies, the effect of the bulk on the brane evolution can be described
through a mirage component, termed generalized dark radiation, in the effective
four-dimensional field equations. Using the covariant formalism, we derive the
exact form of these equations. We also derive an effective conservation
equation involving the brane matter and the generalized dark radiation. At low
energies the coupled brane-bulk system has a purely four-dimensional
description. The applications of the formalism include generalizations of the
Starobinsky model and the Dvali-Gabadadze-Porrati cosmology.
