Delay-tolerant networks (DTNs) are characterized by a possible absence of end-to-end communication routes at any instant. Yet, connectivity can be achieved over time and space, leading to evaluate a given route both in terms of topological length or temporal length. The problem of measuring temporal distances in a social network was recently addressed through post-processing contact traces like email datasets, in which all contacts are punctual in time (i.e., they have no duration). We focus on the distributed version of this problem and address the more general case that contacts can have arbitrary durations (i.e., be non-punctual). Precisely, we ask whether each node in a network can track in real-time how "out-of-date" it is with respect to every other. Although relatively straightforward with punctual contacts, this problem is substantially more complex with arbitrariry long contacts: consecutive hops of an optimal route may either be disconnected (intermittent connectedness of DTNs) or connected (i.e., the presence of links overlap in time, implying a continuum of path opportunities). The problem is further complicated (and yet, more realistic) by the fact that we address continuous-time systems and non-negligible message latencies (time to propagate a single message over a single link), however this latency is assumed fixed and known. We demonstrate the problem is solvable in this general context by generalizing a time-measurement vector clock construct to the case of "non-punctual" causality, which results in a tool we call T-CLOCKS, of independent interest. The rest of the paper shows how T-CLOCKS can be leveraged to solve concrete problems such as learning foremost broadcast trees, network backbones, or fastest broadcast trees in periodic DTNs.