Edge time series decompose interregional correlations (functional connectivity; FC) into their time-varying contributions. Previous studies have revealed that brief, high-amplitude, and globally-defined "events" contribute disproportionately to the time-averaged FC pattern. This whole-brain view prioritizes systems that occupy vast neocortical territory, possibly obscuring extremely high-amplitude co-fluctuations that are localized to smaller brain systems. Here, we investigate local events detected at the system level, assessing their independent contributions to global events and characterizing their repertoire during resting-state and movie-watching scans. We find that, as expected, global events are more likely to occur when large brain systems exhibit events. Next, we study the co-fluctuation patterns that coincide with system events--i.e. events detected locally based on the behavior of individual brain systems. We find that although each system exhibits a distinct co-fluctuation pattern that is dissimilar from those associated with global events, the patterns can nonetheless be grouped into two broad categories, corresponding to events that coincide with sensorimotor and attention systems and, separately, association systems. We then investigate system-level events during movie-watching, discovering that the timing of events in sensorimotor and attention systems decouple, yielding reductions in co-fluctuation amplitude. Next, we show that by associating each edge with its most similar system-averaged edge time series, we recover overlapping community structure, obviating the need for applying clustering algorithms to high-dimensional edge time series. Finally, we focus on cortical responses to system-level events in subcortical areas and the cerebellum. We show that these structures coincide with spatially distributed cortical co-fluctuations, centered on prefrontal and somatosensory systems. Collectively, the findings presented here help clarify the relative contributions of large and small systems to global events, as well as their independent behavior.