The Vehicle Routing Problem with pickups, deliveries and spatiotemporal service constraints ($VRPPDSTC$) is a quite challenging algorithmic problem that can be dealt with in either an offline or an online fashion. In this work, we focus on a generalization, called $VRPPDSTCtd$, in which the travel-time metric is \emph{time-dependent}: the traversal-time per road segment (represented as a directed arc) is determined by some function of the departure-time from its tail towards its head. Time-dependence makes things much more complicated, even for the simpler problem of computing earliest-arrival-time paths which is a crucial subroutine to be solved (numerous times) by $VRPPDSTCtd$ schedulers. We propose two \emph{online} schedulers of requests to workers, one which is a time-dependent variant of the classical Plain-Insertion heuristic, and an extension of it trying to digest some sort of forecasts for future demands for service. We enrich these two online schedulers with two additional heuristics, one targeting for distance-balanced assignments of work loads to the workers and another that makes local-search-improvements to the produced solutions. We conduct a careful experimental evaluation of the proposed algorithms on a real-world instance, with or without these heuristics, and compare their quality with human-curated assignments provided by professional experts (human operators at actual pickup-and-delivery control centers), and also with feasible solutions constructed from a relaxed MILP formulation of $VRPPDSTCtd$, which is also introduced in this paper. Our findings are quite encouraging, demonstrating that the proposed algorithms produce solutions which (i) are significant improvements over the human-curated assignments, and (ii) have overall quality pretty close to that of the (extremely time-consuming) solutions provided by an exact solver for the MILP formulation.