A Conditional Simple Temporal Network (CSTN) includes time-points, temporal constraints, and observation time-points, whose execution yields information during run-time. Time-points and constraints in a CSTN may only apply in certain scenarios. A CSTN is dynamically consistent (DC) if it has a strategy for executing its time-points such that all relevant constraints will be satisfied no matter how the observations turn out. A dynamic strategy can react to observations in real time, but only after arbitrarily small, but positive delays. Recent work introduced a more realistic epsilon-DC property which, for a fixed epsilon>0, requires all reaction times to be bounded below by epsilon. That work presented an exponential algorithm for checking the epsilon-DC property by translating an exponential number of component networks into a Hyper Temporal Network. But it has not yet been implemented or empirically evaluated. This paper begins by presenting an alternative, equivalent semantics for epsilon-dynamic consistency. It then presents a sound-and-complete epsilon-DC-checking algorithm based on the propagation of labeled constraints. Finally, it presents an empirical evaluation of the new algorithm, the first empirical evaluation of any epsilon-DC-checking algorithm in the literature.