A very natural question that arises in the context of stability properties of packet-switched networks is how the dynamic changing of the network link slowdowns precisely affects these properties. In this work, we embark on a systematic study of this question adopting the Adversarial, Quasi- Static Slowdown Queueing Theory model, where an adversary controls the rates of packet injections, determines packet paths and manipulates link slowdowns. Within this model, we show that a network that uses the LIS (Longestin- System) protocol for contention-resolution can be unstable for arbitrarily low injection rates. Moreover, we present involved combinatorial constructions of executions that improve the state-of-the-art instability bound induced by certain known forbidden subgraphs on networks running a certain greedy protocol. These bounds are lower than their known counterparts for the classical Adversarial Queueing Theory model and other dynamic adversarial models.