We study cosmological perturbations in general inflation models with multiple scalar fields and arbitrary kinetic terms, with special emphasis on the multi-field extension of Dirac-Born-Infeld (DBI) inflation. We compute the second-order action governing the dynamics of linear perturbations in the most general case. Specializing to DBI, we show that the adiabatic and entropy modes propagate with a {\it common} effective sound speed and are thus amplified at sound horizon crossing. In the small sound speed limit, we find that the amplitude of the entropy modes is much higher than that of the adiabatic modes. We also derive, in the general case, the third-order action which is useful for studying primordial non-Gaussianities generated during inflation. In the DBI case, we compute the dominant contributions to non-Gaussianities, which depend on both the adiabatic and entropy modes.