This paper addresses the distributed adaptive event-triggered H∞ filtering problem for a class of sector-bounded nonlinear system over a filtering network with time-varying and switching topology. Both topology switching and adaptive event-triggered mechanisms (AETMs) between filters are simultaneously considered in the filtering network design. The communication topology evolves over time, which is assumed to be subject to a nonhomogeneous Markov chain. In consideration of the limited network bandwidth, AETMs have been used in the information transmission from the sensor to the filter as well as the information exchange among filters. The proposed AETM is characterized by introducing the dynamic threshold parameter, which provides benefits in data scheduling. Moreover, the gain of the correction term in the adaptive rule varies directly with the estimation error and inversely with the transmission error. The switching filtering network is modeled by a Markov jump nonlinear system. The stochastic Markov stability theory and linear matrix inequality techniques are exploited to establish the existence of the filtering network and further derive the filter parameters. A co-design algorithm for determining H∞ filters and the event parameters is developed. Finally, some simulation results on a continuous stirred tank reactor and a numerical example are presented to show the applicability of the obtained results.