The interpretation is similar to the diffusive dynamics in a sense that FP description of RD systems likewise operates with a probability distribution which evolves in time due to fluxes brought about by diffusion and reactions. The nature of these fluxes need to be reflected in the FP operator. Also just like in the case of ordinary diffusion the validity of FP picture for RD systems rests on the plausibility of spatial and temporal coarse graining. E.g one demands that 1) system is very large so that one can approximate change of otherwise discrete molecular numbers with differentials of smooth and continuously changing probability distributions and 2) Reactions are of Markovian nature, meaning that thermal motion erases the microscopic dynamics fairly quickly relative to the reactive events and diffusion.
I would say the best source to read up on this topic is Van Kampen's book (Stochastic Processes in Physics and Chemistry) which is the standard citing reference in many research papers and books.