We present a microscopic theory of heat and particle transport of an interacting, low-temperature Bose- Einstein condensate in a quantum point contact.We show that, in contrast to charged, fermionic superconductors, bosonic systems feature tunneling processes of condensate elements, leading to the presence of odd-order harmonics in the AC Josephson current. A crucial role is played by an anomalous tunneling process where condensate elements are coherently converted into phonon excitations, leading to even-order harmonics in the AC currents as well as a DC contribution. At a low bias, we find dissipative components obeying Ohm’s law and bias-independent nondissipative components, in sharp contrast to fermionic superconductors. Analyzing the DC contribution, we find zero thermopower and Lorenz number at zero temperature, a breakdown of the bosonic Wiedemann-Franz law. These results highlight the importance of the anomalous tunneling process inherent to charge-neutral superfluids. The consequences could readily be observed in existing cold-atom transport setups.