The non-equilibrium dynamics of the bond-diluted and the ±J Ising model is studied numerically in two dimensions (2d). In both cases we find evidence for 'blocking'. For strong bond-dilution (0 ≤ p ≤ 0.6) we find that the residual persistence decays exponentially to zero at large times. The 'blocking probability' increases monotonically from zero for p = 0 with increasing bond concentration to a maximum value of approximately 0.46 for p = 0.5 and then decreases to zero for p = 1, the pure case. For the ±J model the residual persistence decays algebraically, just as in the pure model, but with a different persistence exponent. The 'blocking probability' once again exhibits non-monotonic behaviour. Our results suggest that persistence in disordered and pure systems falls into different classes. Furthermore, its behaviour would also appear to depend crucially on the type of disorder present.