In this article, we focus on the systemic expected shortfall and marginal expected shortfall in a multivariate continuous-time risk model with a general càdlàg process. Additionally, we conduct our study under a mild moment condition that is easily satisfied when the general càdlàg process is determined by some important investment return processes. In the presence of heavy tails, we derive asymptotic formulas for the systemic expected shortfall and marginal expected shortfall under the framework that includes wide dependence structures among losses, covering pairwise strong quasi-asymptotic independence and multivariate regular variation. Our results quantify how the general càdlàg process, heavy-tailed property of losses, and dependence structures influence the systemic expected shortfall and marginal expected shortfall. To discuss the interplay of dependence structures and heavy-tailedness, we apply an explicit order 3.0 weak scheme to estimate the expectations related to the general càdlàg process. This enables us to validate the moment condition from a numerical perspective and perform numerical studies. Our numerical studies reveal that the asymptotic dependence structure has a significant impact on the systemic expected shortfall and marginal expected shortfall, but heavy-tailedness has a more pronounced effect than the asymptotic dependence structure.