Classical diffusion across magnetic fields driven by density gradients in hot plasmas is a problem that has been considered by many authors because of its application to many areas in plasma physics. In most cases, however, only particle diffusion in one-temperature plasmas has been considered. Even when the interacting species are allowed to have unequal temperatures, the energy diffusion resulting from the density gradients is not calculated. There are disagreements among existing results, even in the case of single-temperature particle diffusion. Expressions for classical particle and energy diffusion across magnetic fields for multitemperature plasmas are derived from basic principles. The results are then compared with those most often quoted in the literature.