Processes of melt generation and transport beneath back-arc spreading centres are controlled by two endmember mechanisms: decompression melting similar to that at mid-ocean ridges and flux melting resembling that beneath arcs. The Lau Basin, with an abundance of spreading ridges at different distances from the subduction zone, provides an opportunity to distinguish the effects of these two different melting processes on magma production and crust formation. Here we present constraints on the three-dimensional distribution of partial melt inferred from seismic velocities obtained from Rayleigh wave tomography using land and ocean-bottom seismographs. Low seismic velocities beneath the Central Lau Spreading Centre and the northern Eastern Lau Spreading Centre extend deeper and westwards into the back-arc, suggesting that these spreading centres are fed by melting along upwelling zones from the west, and helping to explain geochemical differences with the Valu Fa Ridge to the south, which has no distinct deep low-seismic-velocity anomalies. A region of low S-wave velocity, interpreted as resulting from high melt content, is imaged in the mantle wedge beneath the Central Lau Spreading Centre and the northeastern Lau Basin, even where no active spreading centre currently exists. This low-seismic-velocity anomaly becomes weaker with distance southward along the Eastern Lau Spreading Centre and the Valu Fa Ridge, in contrast to the inferred increase in magmatic productivity. We propose that the anomaly variations result from changes in the efficiency of melt extraction, with the decrease in melt to the south correlating with increased fractional melting and higher water content in the magma. Water released from the slab may greatly reduce the melt viscosity or increase grain size, or both, thereby facilitating melt transport.