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We experimentally investigate the electronic transport properties of a three-terminal graphene Josephson junction. We find that self-heating effects strongly influence the behaviour of this multiterminal Josephson junction (MTJJ) system. We show that existing simulation methods based on resistively and capacitively shunted Josephson junction networks can be significantly improved by taking into account these heating effects. We also investigate the phase dynamics in our MTJJ by measuring its switching current distribution and find correlated switching events in different junctions. We show that the switching dynamics is governed by phase diffusion at low temperatures. Furthermore, we find that self-heating introduces additional damping which results in overdamped I-V characteristics when normal and supercurrents coexist in the device.