In this paper we initiate the generalization of the Adversarial Queueing Theory (aqt) model to capture the dynamics of continuous scenarios in which the usually assumed synchronicity of the evolution is not required anymore. We propose an asynchronous model, named continuous aqt (caqt), in which packets can have arbitrary lengths, and the network links may have different speeds (or bandwidths) and propagation delays. With respect to the standard aqt model, these new features turn out to be significant for the stability of packet scheduling policies that take them into account, but not so much for the stability of networks. From the network point of view, we show that networks with directed acyclic topologies are universally stable, i.e., stable independently of the scheduling policies and traffic patterns used in it. Interestingly enough, this even holds for traffic patterns that make links to be fully loaded. Finally, it turns out that the set of universally stable networks remains the same as in the aqt model and, therefore, the property of universal stability of networks is decidable in polynomial time. Concerning packet scheduling policies, we show that the well-known lis, sis, ftgand nfsscheduling policies remain universally stable in the caqt model. We introduce other scheduling policies that, although being universally stable in the aqt model, they are unstable under the caqt model.