Gene drives are genetic constructs that can spread deleterious alleles with potential application to population suppression of harmful species. As gene drives can potentially spill over to other populations or species, control measures and fail-safe strategies must be considered. Gene drives can generate a rapid change in the population's genetic composition, leading to substantial demographic decline, processes that are expected to occur at a similar timescale during gene drive spread. We developed a gene drive model that combines evolutionary and demographic dynamics in a two-population setting. The model demonstrates how feedback between these dynamics generates additional outcomes to those generated by the evolutionary dynamics alone. We identify an outcome of particular interest where short-term suppression of the target population is followed by gene swamping and loss of the gene drive. This outcome can prevent spillover and is robust to the evolution of resistance, suggesting it may be suitable as a fail-safe strategy for gene drive deployment.