An extension to rarefied flow regimes of the lattice Boltzmann method-based model for miscible mixtures developed by Vienne et al. (Physical Review E 100.2 (2019): 023309) is presented. The model is applied to study the gas phase separation phenomenon in mixture flows that traditional macroscopic approaches fail to predict. The extension includes a wall function approach with an empirical coefficient to define an effective mean free path in solid geometries, which locally defines the kinematic viscosity and binary diffusion coefficients. The algorithm is also modified by a local multi-relaxation time collision operator and slip boundary conditions at solid walls. Gaseous mixture flow simulations are conducted through a 2D plane microchannel within the slip and early transition flow regimes. Despite a miscible gaseous phase, the mixture loses its homogeneity and independent velocity profiles for each component are observed in the rarefied regime and captured with the current modeling. In addition, the gas separation phenomenon increases with the rarefaction rate and the molecular mass ratio. The individual treatment of the species within mixture flows in the developed lattice Boltzmann model helps understanding the increasing independent behavior of the individual species within the mixture as the regime becomes more rarefied.