We study nonequilibrium (NE) transport in four-terminal (three-terminal) topological superconductor (SC)-quantum dot (QD) topological superconductor junctions, where the QD is connected via tunneling barriers to the two TS leads and two (one) normal leads (N), respectively. For the four-terminal junction, we find that when increasing the Zeeman field from 0 to a critical value, the supercurrent profile evolves from a typical s-wave pattern to a pure p-wave pattern. In addition, by analyzing the zero-phase difference supercurrent as a function of voltage [Formula: see text] applied to the normal leads and the Zeeman field h applied to the SC, the low-momentum gap [Formula: see text] can be inferred by utilizing the fact that the emergence of a tunneling-induced current should be satisfied under the condition [Formula: see text]. For the three-terminal junction, the NE supercurrent can reveal the quasi-Andreev bound state by exploiting the Andreev reflection process-induced current occurring between the N and SC. Our findings provide an arguably easier route for manifesting the topological phase transition by observing the gap collapse and then reopening as the Zeeman field increases through multi-terminal NE transport.