Different integral representations for the mass flux of inertial particles transported by turbulent gas flows have been proposed. These are discussed and analyzed. Each formulation provides its own insights into the underlying physical processes governing the resulting flux. However, none of the representations, as it stands, provides an explicit closed-form expression in terms of known statistical properties of the flow and parameters governing particle dynamics. We consider the representations in terms of their potential for reduction to closed-form models. To enable an analysis uncomplicated by the presence of many coupled interactions, we confine our attention to the classic test case of monodisperse particles in homogeneous, isotropic turbulent flows, and subject to a uniform gravitational field. The modification of the mean particle settling velocity resulting from their preferential sampling of fluid velocities is captured by the flux representations. A distribution-based symmetry analysis coupled with a correlation splitting technique is used to reduce and simplify the terms appearing in the flux integrals. This prompts a strategy for closure modeling of the resulting expressions in terms of correlations between the sampled fluid velocity and fluid strain-rate fields. Results from particle-trajectory-based simulations are presented to assess the potential of this closure strategy.