During solar maximum, interplanetary coronal mass ejections and associated interplanetary shocks occur frequently. These structures are often accompanied by high levels of low-frequency compressive wave turbulence, which requires a nonlinear extension of standard quasi-linear theory to properly describe energetic particle transport in their vicinities. The same might be true for the solar wind termination shock. We present a nonlinear diffusive kinetic theory for suprathermal particle transport and stochastic acceleration along the background magnetic field in strong compressive dynamic wave turbulence to which small-scale Alfven waves are coupled. Our theory shows that the standard cosmic-ray transport equation must be revised for low suprathermal particle energies to accommodate fundamental changes in spatial diffusion (standard diffusion becomes turbulent diffusion), as well as modifications to particle convection and adiabatic energy changes. In addition, a momentum diffusion term, which generates accelerated suprathermal particle spectra with a hard power law, must be added. Such effective first-stage acceleration possibly leads to efficient injection of particles into second-stage diffusive shock acceleration, as described by standard theory.