The turbulent energy fluxes, including up-gradient 'energy pinch' effects, are derived using the nonlinear bounce-kinetic equation for trapped electrons and the nonlinear gyrokinetic equation for ions in toroidal geometry. The quasi-universal type of inward turbulent equipartition (TEP) energy pinch is recovered for both ions and trapped electrons, with different field dependence coefficients due to toroidal effects. A contribution from the density gradient to an outward convective energy flux is also obtained. The direction of the total energy convection is primarily determined by the competition between the TEP energy pinch and the outward density gradient driven energy convection. The magnetic shear dependence of the electron energy pinch is discussed. The energy pinches can provide possible explanations for some puzzling experimental observations.