Motivated by the emerging applications of liquid-infused surfaces (LIS), we study the drag reduction and robustness of transverse flows over two-dimensional microcavities partially filled with an oily lubricant. Using separate simulations at different scales, characteristic contact line velocities at the fluid-solid intersection are first extracted from nano-scale phase field simulations and then applied to micron-scale two-phase flows, thus introducing a multiscale numerical framework to model the interface displacement and deformation within the cavities. As we explore the various effects of the lubricant-to-outer-fluid viscosity ratioμ 2 /μ 1 , the capillary number Ca, the static contact angle θ s , and the filling fraction of the cavity δ, we find that the effective slip is most sensitive to the parameter δ. The effects ofμ 2 /μ 1 and θ s are generally intertwined, but weakened if δ < 1. Moreover, for an initial filling fraction δ = 0.94, our results show that the effective slip is nearly independent of the capillary number, when it is small. Further increasing Ca to about 0.01μ 1 /μ 2 , we identify a possible failure mode, associated with lubricants draining from the LIS, forμ 2 /μ 1 0.1. Very viscous lubricants (e.g.μ 2 /μ 1 > 1), on the other hand, are immune to such failure due to their generally larger contact line velocity.