Active droplets, membraneless compartments driven by internal chemical reactions, are compelling models for protocells and synthetic life. A central challenge is to program their dynamic behaviors using heritable genetic information, which would grant them the capacity to evolve. Here, we create transiently active RNA droplets by integrating sites for ribozyme catalysis directly into the sequence of self-assembling, four-arm RNA nanostars. To enable perfusion and observe the resulting dynamics over time, we develop a method for trapping individual droplets in hydrogel cages by targeted in situ photopolymerization. This enables us to quantify the sequence-programmable droplet dissolution and to control the degradation kinetics by choosing between fast (hammerhead) and slow (hairpin) ribozymes. Furthermore, we trigger the segregation of mixed droplet populations via the sequence-specific cleavage of a chimeric linker RNA. The droplet-encapsulated DNA templates code for the regrowth of new droplets, establishing the proof-of-principle for a minimal, genetically encoded cycle of dissolution and regrowth. By directly linking RNA sequence to droplet stability, composition, and life-cycle dynamics, our work provides a robust platform for engineering evolvable materials and advancing the bottom-up construction of synthetic cells.