We study the coupled two-species non-equilibrium reaction-controlled diffusion model introduced by Trimper et al. [Phys. Rev. E 62, 6071 (2000)] by means of detailed Monte Carlo simulations in one and two dimensions. Particles of type A may independently hop to an adjacent lattice site provided it is occupied by at least one B particle. The B particle species undergoes diffusion-limited reactions. In an active state with nonzero, essentially homogeneous B particle saturation density, the A species displays normal diffusion. In an inactive, absorbing phase with exponentially decaying B density, the A particles become localized. In situations with algebraic decay ρB(t) ∼ t^−∞B, as occuring either at a non-equilibrium continuous phase transition separating active and absorbing states, or in a power-law inactive phase, the A particles propagate subdiffusively with mean-square displacement ⟨^→x(t)²A⟩~t^1-∞A. We find that within the accuracy of our simulation data, αA ≈ αB as predicted by a simple mean-field approach. This remains true even in the presence of strong spatio-temporal fluctuations of the B density. However, in contrast with the mean-field results, our data yield a distinctly non-Gaussian A particle displacement distribution n_A(^→x, t) that obeys dynamic scaling and looks remarkably similar for the different processes investigated here. Fluctuations of effective diffusion rates cause a marked enhancement of n_A(^→x, t) at low displacements |^→x|, indicating a considerable fraction of practically localized A particles, as well as at large traversed distances.