The problem of scheduling moldable tasks has been widely studied, in particular when tasks have dependencies (i.e., task graphs), or when tasks are released on-the-fly (i.e., online). However, few study has focused on both (i.e., online scheduling of moldable task graphs). In this paper, we derive constant competitive ratios for this problem under several common yet realistic speedup models for the tasks (roofline, communication, Amdahl, and a combination of them). We also provide the first lower bound on the competitive ratio of any deterministic online algorithm for arbitrary speedup model, which is not constant but depends on the number of tasks in the longest path of the graph.