This article introduces a novel methodology, Network Simulator-centric Compositional Testing (NSCT), to enhance the verification of network protocols with a particular focus on time-varying network properties. NSCT follows a Model-Based Testing (MBT) approach. These approaches usually struggle to test and represent time-varying network properties. NSCT also aims to achieve more accurate and reproducible protocol testing. It is implemented using the Ivy tool and the Shadow network simulator. This enables online debugging of real protocol implementations. A case study on an implementation of QUIC (picoquic) is presented, revealing an error in its compliance with a time-varying specification. This error has subsequently been rectified, highlighting NSCT's effectiveness in uncovering and addressing real-world protocol implementation issues. The article underscores NSCT's potential in advancing protocol testing methodologies, offering a notable contribution to the field of network protocol verification.