We propose a space-time Finite Volume scheme on moving Chimera grids for a general advection-diffusion problem. Special care is devoted to grid overlapping zones in order to devise a compact and accurate discretization stencil to exchange information between different mesh patches. Like in the ADER method, the equations are discretized on a space-time slab. Thus, instead of time-dependent spatial transmission conditions between relatively moving grid blocks, we define interpolation polynomials on arbitrarily intersecting space-time cells at the block boundaries. \rtwo{Through this scheme, a mesh-free FEM-predictor/FVM-corrector approach is employed for representing the solution.} In this discretization framework, a new space-time Local Lax-Friederichs (LLF) stabilization speed is defined by considering both the advective and diffusive nature of the equation. The numerical illustrations for linear and non-linear systems show that background and foreground moving meshes do not introduce spurious perturbation to the solution, uniformly reaching second order accuracy in space and time. Finally, it is shown that several foreground meshes, possibly overlapping and with independent displacements, can be employed thanks to this approach
