The introgression of genetic traits through gene drive may serve as a powerful and widely applicable method of biological control. However, for many applications, a self-perpetuating gene drive that can spread beyond the specific target population may be undesirable and preclude use. Daisy-chain gene drives have been proposed as a means of tuning the invasiveness of a gene drive, allowing it to spread efficiently into the target population, but be self-limiting beyond that. Daisy-chain gene drives are made up of multiple independent drive elements, where each element, except one, biases the inheritance of another, forming a chain. Under ideal inheritance biasing conditions, the released drive elements remain linked in the same configuration, generating copies of most of their elements except for the last remaining link in the chain. Through mathematical modelling of populations connected by migration, we have evaluated the effect of resistance alleles, different fitness costs, reduction in the cut-rate, and maternal deposition on two alternative daisy-chain gene drive designs. We find that the self-limiting nature of daisy-chain gene drives makes their spread highly dependent on the efficiency and fidelity of the inheritance biasing mechanism. In particular, reductions in the cut-rate and the formation of non-lethal resistance alleles can cause drive elements to lose their linked configuration. This severely reduces the invasiveness of the drives and allows for phantom cutting, where an upstream drive element cuts a downstream target locus despite the corresponding drive element being absent, creating and biasing the inheritance of additional resistance alleles. This phantom cutting can be mitigated by an alternative indirect daisy-chain design. We further find that while dominant fitness costs and maternal deposition reduce daisy-chain invasiveness, if overcome with an increased release frequency, they can reduce the spread of the drive into a neighbouring population.