We present a numerical method for simulating mixtures of rarefied gases that interact with moving boundaries and rigid bodies. The gas mixture is described by the general consistent BGK model for gas mixtures (Bobylev et al. in Kinetic Related Models 11(6):1377–1393, 2018) [1]. The BGK equations are solved using an Arbitrary Lagrangian-Eulerian method, in which grid points move with the local mean velocity of the gas (Tiwari et al. in J Comput Phys 458, 2022) [2]. The main advantage of the moving grid is that the algorithm can deal well with cases where the domain boundaries are time-dependent and the simulation domain contains rigid objects. Due to the irregular nature of the grid, we use a novel MUSCL-like Moving Least Squares Method (MLS) spatial discretization coupled with a second-order IMEX method. To avoid spurious oscillations at discontinuities, we use the so-called MOOD algorithm (Clain et al. in J Comput Phys 230(10):4028–4050, 2011)  [3]. As a proof-of-concept, we apply the algorithm to Sod’s shock tube and a moving boundary problem.