Numerical studies of lattice-gas models are well suited to describe multiadsorbate systems. One example is the underpotential deposition of Cu on Au(111) in the presence of sulfuric acid. Preliminary results from dynamic Monte Carlo simulations of the evolution of the adsorbed layer during potential-step experiments across phase transitions are presented for this particular system. The simulated current profiles reproduce a strong asymmetry seen in recent experiments. Examination of the microscopic structures that occur during the simulated evolution processes raises questions that need to be investigated by further experimental and theoretical study.