The kinetic theory is formulated with respect to anholonomic frames of reference on curved spacetimes. By using the concept of nonlinear connection we develop an approach to modelling locally anisotropic kinetic processes and, in corresponding limits, the relativistic nonequilibrium thermodynamics with local anisotropy. This leads to a unified formulation of the kinetic equations on (pseudo) Riemannian spaces and in various higher dimensional models of Kaluza-Klein type and/or generalized Lagrange and Finsler spaces. The transition rate considered for the locally anisotropic transport equations is related to the differential cross section and spacetime parameters of anisotropy. The equations of states for pressure and energy in locally anisotropic thermodynamics are derived. The obtained general expressions for heat conductivity, shear, and volume viscosity coefficients are applied to determine the transport coefficients of cosmic fluids in spacetimes with generic local anisotropy. We emphasize that such local anisotropic structures are induced also in general relativity if we are modelling physical processes with respect to frames with mixed sets of holonomic and anholonomic basis vectors which naturally admits an associated nonlinear connection structure.