We present a numerical method for simulating rarefied gases that interact with moving boundaries and rigid bodies. The rarefied gas is modelled by the BGK-Boltzmann equation written in Lagrangian form, and the motion of the rigid bodies is governed by the Newton-Euler equations. The equation is then solved using an Arbitrary Lagrangian-Eulerian (ALE) method in which the grid points move with the local mean velocity of the gas. The main advantage of the moving grid is that the algorithm can deal well with cases where the simulation domain contains rigid objects, since no cut-cell technique or immersed boundaries are required. In addition, the CFL restriction in the ALE formulation is much less severe than classical Eulerian schemes, especially for high Mach number flows. The BGK equations are solved using higher-order Implicit-Explicit (IMEX) Runge-Kutta methods in which the advection term is solved explicitly, and the stiff relaxation terms implicitly. The main novelty of our approach is the use of a meshless MUSCL-like spatial discretization, in which the states are reconstructed at the midpoints using a Moving Least Squares (MLS) method. This approach provides a general framework for the stable, high-order discretisation of hyperbolic PDEs using meshfree methods. Since only a first-order MLS method can guarantee the maximum principle, we make use of a Multi-dimensional Optimal Order Detection (MOOD) scheme to avoid spurious oscillations due to the transport step. We compare our approach to a WENO MLS method. To showcase the versatility of the algorithm, we apply it to several test cases in 1D and 2D. In 1D, we use the new second and fourth-order methods for Sod's shock tube and the moving slider problem. For Sod's shock tube, our methods are significantly less diffusive than the WENO MLS method. In 2D, we consider the driven cavity problem with moving objects inside the domain, and the shear layer problem. Both simulations are performed with an ARS222 IMEX scheme combined with a second-order meshless MUSCL scheme and MOOD on 32 CPUs. The combination of the MOOD with the versatile MLS scheme results in a promising algorithm for complex simulations in the field of rarefied gas dynamics.