In the forced flow of a gaseous mixture in a capillary or a porous medium in a field of partial-pressure gradients, a number of effects occur (the diffusion baroeffect [1, 2], the mixture-separation effect [3, 4], etc.), a rigorous analysis of which requires the inclusion of Boltzmann's kinetic equation. The main object of the kinetic consideration in this case is to obtain expressions for the flows of the mixture components, averaged over the cross section of the channel or referred to unit surface of the porous medium. This problem has been solved in a number of papers [5–7] for channels of correct geometry (a plane slit or a circular cylindrical capillary) using the linearized kinetic equation with the model BGK integral of the collisions in the Hamel form [8]. In [9] the flow of a mixture in a plane channel was considered using the accurate linearized collison operator, but subsequent use of the moment method of solution was confined to the solid-sphere model of the molecules. The limitation of the models used does not enable the accuracy of the results obtained to be guaranteed, particularly with regard to such kinetic quantities as the diffusion slip coefficient or the barodiffusion constant of the gaseous mixture in the channel. It is well known, in particular [8], that no matter how the parameters of the slip in the BGK model for the mixture are chosen, it is not possible to ensure an adequate description of the diffusion and the viscosity of the mixture simultaneously even for normal hydrodynamic flow. Below we solve the problem of the flow of a mixture in a channel using the linearized kinetic equation with the collision operator in the model form proposed by McCormack [10]. The advantage of this model, based on the equivalence of the N-order moments of the accurate and the model integrals, is the fact that it automatically gives a correct description of the mixture in the hydrodynamic limit for an arbitrary interaction between the molecules. For the case of isothermal flow of the mixture considered here, satisfactory approximation is obtained for N=2, when the diffusion velocity and the partial tensor of the viscous stresses are defined by the moments of the distribution function, in addition to the density, the mean-mass velocity, and the temperature. The method of solution used is limited to the region small Khudsen numbers (Kn=λ/d, where λ is the effective mean free path of the molecules and d is the characteristic transverse dimension of the channel). In this case the region of mixture flow in the channel can be divided into two: the region far from the walls where the usual hydrodynamic approximations hold, and a thin region close to the walls (the Knudsen layer), the solution in which must be considered taking into account the true boundary conditions for the distribution function on the walls of the channel and the asymptotic conditions on the external boundary of the layer. The solution of the problem in the external region gives an expression for the velocities of each of the components of the mixture in the channel with certain fictional macroscopic boundary conditions on its walls. These conditions are found by solving the kinetic equation in the Knudsen layer, for which the method of total moments is used. This formulation of the problem is closely connected with the problem of determining the diffusion slip velocity of a mixture [11–14], in view of the fact that the consideration begins with precisely this case. The problem is then generalized to the case of the flow of a mixture in plane and cylindrical channels when there are both concentration gradients and a gradient of the total pressure of the mixture present. It is known in this case, in particular, that the barodiffusion constant in the expression for the difference in the velocity components averaged over the cross section is equal to (with opposite sign) the diffusion slip coefficient, which agrees with the general conclusions of the thermodynamics of irreversible processes [15].