Parallel execution of simulation runs has become indispensable in different research areas recently. One of the most promising and powerful models in science are cellular automata (CA). This paper describes the approach to migrate parallel execution of a cellular automata simulation program called ParCASim via grid-capable message passing from cluster to grid environments. The applicability of the program is demonstrated for the calculation of sophisticated single-file diffusion problems and exploits the improved compute power of the underlying grid in order to deliver results faster and more precise than on classic cluster architecture. In this paper, we discuss ParCASim as a tool, and the utilised cluster-to-cluster interconnection is outlined. Furthermore, the paper describes single-file diffusion and how it is calculated via the CA model. Finally, time and efficiency measurements are taken which show the applicability of the approach.