Waste management networks are essential for environmental protection and public health, but vulnerable to regulatory evasion, fraud, and illegal trading. Detecting potentially illicit activities in the network requires robust anomaly detection systems. However, the complexity of interactions between heterogeneous entities such as recycling companies, individuals, and other organisations combined with temporal irregularities and network dynamics makes conventional fraud detection approaches less effective. In this study, we introduce a dynamic graph-based framework that combines statistical change detection methods, the Page-Hinkley and CUSUM tests, alongside a deep learning model, LSTM-VAE, to detect suspicious activities in the Portuguese waste management network. Using real-world waste transfer records, we engineered temporal and network features to reveal a wide range of anomalies, including abrupt shifts in activity and unusual connectivity patterns, such as those involving collusive triangles. The evaluation was based on four pre-labeled anomalous companies identified by regulators. Our results show that while individual methods excel at detecting certain behaviors, their combination provides robust coverage of diverse anomaly types, with each anomalous company identified by at least three techniques. This approach demonstrates the importance of integrating temporal and network-based analysis, offering regulatory authorities a scalable tool to prioritize inspections, enhancing accountability in the waste management network.