Polymer covered emulsion droplets have a considerable number of applications ranging from active cosmetics to advance drug delivery systems. In many of these systems the emulsion droplets do not exist in isolation but interact with other drops, surfaces and particles. In a step towards understanding how these complex mechanical interactions take place, we examine the interaction between a block copolymer covered emulsion droplet (polystyrene-b-poly(ethylene oxide) (PS-PEO) covered toluene) and a flat mica interface. As buoyancy pushes the droplet upwards, it buckles in as it nears the mica and traps a droplet of the surrounding fluid. The trapped outer fluid (water/glycerine in our experiment) drains out through an annular region of PEO brush. This study focuses on the late stage drainage, unique to large molecule surfactants, and examines the effects of the polymer and droplet size on the drainage rate. We introduce a scaling model of the drainage which highlights the importance of three lengthscales in the problem - the brush height, the slip length along the emulsion drop interface and the width of the annular contact region.