We study the kinetics of the reaction front in the A+B→C reaction-diffusion system with reactants initially separated by a semipermeable membrane. The semipermeable membrane allows only one reactant species to go through ("penetrating species") while the other reactant species is sterically prohibited from penetration. Theoretically, the ratio of the diffusive fluxes of the two species has been defined before as a control parameter and it was predicted to give rise to a localization-delocalization transition of the reaction front. In this paper we show the experimental realization of a dynamical localization-delocalization transition, in a system consisting of the reactants Ca2+ and calcium green-1 dextran, separated by a finite-sized cellulose membrane. The dynamical transition results from the continuous change in time of the flux of the penetrating species at the reaction boundary. Here this time-dependent flux is attributed to the free diffusion of the penetrating species through a membrane with a finite thickness. The dynamical transition is exemplified by the kinetic behavior of the front characteristics which exhibits several time regimes - an early time, an intermediate time, and an asymptotic time regime. The crossover times between these regimes are found to depend on the membrane thickness, a parameter not considered before to our knowledge. Monte Carlo simulations show good agreement with the finite-time experiments.