To properly capture the complexity of influence propagation phenomena in real-world contexts, such as those related to viral marketing and misinformation spread, information diffusion models should fulfill a number of requirements. These include accounting for several dynamic aspects in the propagation (e.g., latency, time horizon), dealing with multiple cascades of information that might occur competitively, accounting for the contingencies that lead a user to change her/his adoption of one or alternative information items, and leveraging trust/distrust in the users’ relationships and its effect of influence on the users’ decisions. To the best of our knowledge, no diffusion model unifying all of the above requirements has been developed so far. In this work, we address such a challenge and propose a novel class of diffusion models, inspired by the classic linear threshold model, which are designed to deal with trust-aware, non-competitive as well as competitive time-varying propagation scenarios. Our theoretical inspection of the proposed models unveils important findings on the relations with existing linear threshold models for which properties are known about whether monotonicity and submodularity hold for the corresponding activation function. We also propose strategies for the selection of the initial spreaders of the propagation process, for both non-competitive and competitive influence propagation tasks, whose goal is to mimic contexts of misinformation spread. Our extensive experimental evaluation, which was conducted on publicly available networks and included comparison with competing methods, provides evidence on the meaningfulness and uniqueness of our models.