To my knowledge, conventional Monte Carlo simulations cannot tell you anything about the time-dependent properties of a system. The progression from state to state in, say, Metropolis Monte Carlo has nothing to do with the system's evolution in time.
Let's contrast with equilibrium molecular dynamics simulation. In MD, you actually evolve particles in time according to their microscopic equations of motion. In the course of evolving one time step, you have to ask dynamical questions like, "What is each particles' current velocity?" and "What is the net force acting on each particle?" In contrast, with each Monte Carlo step, you're singling out one particle, teleporting it and asking, "Did this move bring the system closer to or further from equilibrium?" Note my word choice here: "teleporting" in Monte Carlo versus "evolving" in MD.
These are two routes to the same goal: getting accurate estimates for equilibrium ensemble-averaged quantities like the pressure, pair distribution function, or free energy. In Monte Carlo, the averaging happens over several static configurations, each assumed statistically independent from one another. In MD, the averaging happens over a long time interval. If the system under study is ergodic, these two methods should give the same answers.
However, for time-dependent quantities, you have to care about how the system gets from point A to point B. In MD, the reason for making each move is unambiguous: we're just obeying Newton's second law (within some numerical accuracy). In Monte Carlo, you choose small moves based on convenience. No physical law prevents you from trying to totally reshuffle the particles every step; it's just really inefficient because there's a high probability of having to reject that move later on. Experience (and statistical mechanics) shows small moves are better.
In each case, we end up with the situation that sequential steps are correlated with one another. In the MD case, those correlations are meaningful, and we can use them to compute things like diffusion coefficients and viscosities and so on. In the Monte Carlo case, the correlations are artificial; sequential steps are correlated because we designed them to be so.
To see this, let's say you try to extract the mean-square displacement from a Monte Carlo sequence. The answer you get will depend on what you set to be the maximum trial displacement!