In recent years, there has been growing interest in the application of temporal reasoning approaches and non-monotonic logics from artificial intelligence in dynamic systems that generate data. A well-known approach to temporal reasoning is the use of a progression technique, which allows for the online computation of logical consequences of a logical knowledge base over time. We consider a progression technique for Temporal Here and There and Temporal Equilibrium Logic, which is the logic underlying answer programming over linear-temporal logic (LTL). Compared to usual LTL online computation, where the goal is to check whether a trace is compliant with a temporal specification, our approach provides also the means to compute non-monotonic temporal reasoning over a trace of observations. Besides formal notions and results, we also present an algorithm for performing progression to monitor a dynamic system, which has been implemented as a proof of concept and allows for handling expressive application scenarios.