Large amplitude compressional type ULF waves can propagate from the magnetosheath to the magnetopause where there are large gradients in density, pressure and magnetic field. These gradients efficiently couple compressional waves with shear/kinetic Alfv¿n waves near the Alfv¿n fieldline resonance location (&ohgr;=k∥vA). We present a solution of the kinetic-MHD wave equations for this process using a realistic equilibrium profile including full ion Larmor radius effects and wave-particle resonance interactions for electrons and ions to model the dissipation. For northward IMF a KAW propagates backward to the magnetosheath. For southward IMF the wave remains in the magnetopause but can propagate through the k∥=0 location. The quasilinear theory predicts that transport due to KAWs at the magnetopause primarily results from the perpendicular electric field coupling with magnetic drift effects with diffusion coefficient D⊥~109 m2/s. For southward IMF additional transport can occur because magnetic islands form at the k∥=0 location. Due to the broadband nature of the observed waves these islands can overlap leading to stochastic transport which is larger than that due to quasilinear effects.