  We derive a non-equilibrium finite-temperature kinetic theory for a binary
mixture of two interacting atomic Bose-Einstein condensates and use it to
explore the degree of hydrodynamicity attainable in realistic experimental
geometries. Based on the standard separation of timescale argument of kinetic
theory, the dynamics of the condensates of the multi-component system are shown
to be described by dissipative Gross-Pitaevskii equations, self-consistently
coupled to corresponding Quantum Boltzmann equations for the non-condensate
atoms: on top of the usual mean field contributions, our scheme identifies a
total of eight distinct collisional processes, whose dynamical interplay is
expected to be responsible for the systems equilibration. In order to provide
their first characterization, we perform a detailed numerical analysis of the
role of trap frequency and geometry on collisional rates for experimentally
accessible mixtures of $^{87}$Rb-$^{41}$K and $^{87}$Rb-$^{85}$Rb, discussing
the extent to which the system may approach the hydrodynamic regime with regard
to some of those processes, as a guide for future experimental investigations
of ultracold Bose gas mixtures.
