We study aftershock sequences of six major earthquakes in New Zealand, including the 2016 M7.8 Kaikaoura and 2016 M7.1 North Island earthquakes. For Kaikaoura earthquake, we assess the expected number of long-delayed large aftershocks of M5+ and M5.5+ in two periods, 0.5 and 3 years after the main shocks, using 75 days of available data. We compare results with obtained for other sequences using same 75-days period. We estimate the errors by considering a set of magnitude thresholds and corresponding periods of data completeness and consistency. To avoid overestimation of the expected rates of large aftershocks, we presume a break of slope of the magnitude–frequency relation in the aftershock sequences, and compare two models, with and without the break of slope. Comparing estimations to the actual number of long-delayed large aftershocks, we observe, in general, a significant underestimation of their expected number. We can suppose that the long-delayed aftershocks may reflect larger-scale processes, including interaction of faults, that complement an isolated relaxation process. In the spirit of this hypothesis, we search for symptoms of the capacity of the aftershock zone to generate large events months after the major earthquake. We adapt an algorithm EAST, studying statistics of early aftershocks, to the case of secondary aftershocks within aftershock sequences of major earthquakes. In retrospective application to the considered cases, the algorithm demonstrates an ability to detect in advance long-delayed aftershocks both in time and space domains. Application of the EAST algorithm to the 2016 M7.8 Kaikoura earthquake zone indicates that the most likely area for a delayed aftershock of M5.5+ or M6+ is at the northern end of the zone in Cook Strait.