The structure of toroidicity-induced Alfven eigenmodes (TAE) and kinetic TAE (KTAE) with large mode numbers is analysed and the linear power transfer from energetic particles to these modes is calculated in the low-shear limit when each mode is localized near a single gap within an interval whose total width Delta ^out is much smaller than the radius r_m of the mode location. Near its peak where most of the mode energy is concentrated, the mode has an inner scale length Delta ⁱⁿ, which is much smaller than Delta ^out. The scale Delta ⁱⁿ is determined by toroidicity and kinetic effects, which eliminate the singularity of the potential at the resonant surface. This work examines the case when the drift orbit width of energetic particles Delta _b is much larger than the inner scale length Delta ⁱⁿ, but arbitrary compared to the total width of the mode. It is shown that the particle-to-wave linear power transfer is comparable for the TAE and KTAE modes in this case. The ratio of the energetic particle contributions to the energetic particle drive for the TAE and KTAE modes is then roughly equal to the inverse ratio of the mode energies. It is found that in the low-shear limit the energetic particle drive for the KTAE modes can be larger than that for the TAE modes.