Pellet injection is an important means to fuel and control discharges and mitigate disruptions in reactor-scale fusion devices. To accurately assess the efficiency of these applications, it is necessary to account for the drift of the ablated material towards the low-field side. In this study, we have implemented a semi-analytical model for ablation cloud drifts in the numerical disruption modelling tool DREAM. We show that this model is capable of reproducing the density evolution in shattered pellet injection (SPI) experiments in ASDEX Upgrade, for model parameters within the expected range. The model is then used to investigate the prospects for disruption mitigation by staggered SPIs in DT H-mode ITER scenarios. We find that the drifts may decrease the assimilation of pure deuterium SPIs by about an order of magnitude, which may be important to consider when designing the disruption mitigation scheme in ITER. The ITER scenarios studied here generally result in similar multi-MA runaway electron (RE) currents, regardless of the drift assumptions, but the effect of the drift is larger in situations with a fast and early thermal quench. The RE current may also be more strongly affected by the drift losses when accounting for RE losses caused by the vertical plasma motion.