Rapid evolutionary processes can produce drastically different outcomes when studied in panmictic population models versus spatial models where the rate of evolution is limited by dispersal. One such process is gene drive, which allows "selfish" genetic elements to quickly spread through a population. Engineered gene drive systems are being considered as a means for suppressing disease vector populations or invasive species. While laboratory experiments and modeling in panmictic populations have shown that such drives can rapidly eliminate a population, it is not yet clear how well these results translate to natural environments where individuals inhabit a continuous landscape. Using spatially explicit simulations, we show that instead of population elimination, release of a suppression drive can result in what we term "chasing" dynamics. This describes a condition in which wild-type individuals quickly recolonize areas where the drive has locally eliminated the population. Despite the drive subsequently chasing the wild-type allele into these newly re-colonized areas, complete population suppression often fails or is substantially delayed. This delay increases the likelihood that the drive becomes lost or that resistance evolves. We systematically analyze how chasing dynamics are influenced by the type of drive, its efficiency, fitness costs, as well as ecological and demographic factors such as the maximal growth rate of the population, the migration rate, and the level of inbreeding. We find that chasing is generally more common for lower efficiency drives and in populations with low dispersal. However, we further find that some drive mechanisms are substantially more prone to chasing behavior than others. Our results demonstrate that the population dynamics of suppression gene drives are determined by a complex interplay of genetic and ecological factors, highlighting the need for realistic spatial modeling to predict the outcome of drive releases in natural populations.