Realistic models of service networks must consider the evolution of interactions with external systems to evaluate emergent response effects on individual network performance. This paper introduces a new dynamic methodology for the assessment of systemic fragility propagation across interdependent networks subjected to seismic action that improves existing static methodologies. Interdependencies are discrete, unidirectional relationships between elements of distinct networks, which are able to influence response evolution from transient to steady-state stages. Comparisons of systemic fragility curves results for isolated and interdependent power and water networks display the importance of interdependence strength and density properties. For the test water network, inter-systemic failure propagation increases its connectivity loss by up to 24%, while high interdependence strengths make the median fragility rise up to 56.2%. In contrast, reductions of interdependence density improve the median water fragility up to 81.7%. Insights obtained from this model, and its associated sequential fragility algorithm, reveal complex coupling patterns and interdependence-based mitigation strategies that are essential for lifeline system management.