Non-wetting surfaces containing micro/nanotextures impregnated with lubricating liquids have recently been shown to exhibit superior non-wetting performance compared to superhydrophobic surfaces that rely on stable air–liquid interfaces. Here we examine the fundamental physico-chemical hydrodynamics that arise when droplets, immiscible with the lubricant, are placed on and allowed to move along these surfaces. We find that these four-phase systems show novel contact line morphology comprising a finite annular ridge of the lubricant pulled above the surface texture and consequently as many as three distinct 3-phase contact lines. We show that these distinct morphologies not only govern the contact line pinning that controls droplets' initial resistance to movement but also the level of viscous dissipation and hence their sliding velocity once the droplets begin to move.