The most recent deglaciation resulted in a global sea-level rise of some 120 m over ca 12 000 years. A moving boundary numerical model is developed to predict the response of rivers to this rise. The model was motivated by experiments at small scale, which have identified two modes describing the transgression of a river mouth: (i) autoretreat without abandonment of the river delta (no sediment starvation at the topset–foreset break); and (ii) sediment-starved autoretreat with abandonment of the delta. In the latter case, transgression is far more rapid, and its effects are felt much further upstream of the river mouth. A moving boundary numerical model that captures these features in experimental deltas is adapted to describe the response of the Fly–Strickland River system, Papua New Guinea. In the absence of better information, the model is applied to the case of sea-level rise without local climate change in New Guinea. The model suggests that: (i) sea-level rise has forced the river mouth to transgress over 700 km since the last glacial maximum; (ii) sediment-starved autoretreat has forced enough bed aggradation to block a tributary with a low sediment load and create the present-day Lake Murray; (iii) the resulting aggradation was sufficient to move the gravel–sand transition on the Strickland River upstream; (iv) the present-day Fly Estuary may be, in part, a relict river valley drowned by sea-level rise and partially filled by tidal effects; and (v) the Fly River is presently reforming its bankfull geometry and prograding into the Fly Estuary. A parametric study with the model indicates that sediment concentration during floods plays a key role in determining whether or not, and to what extent, transgression is expressed in terms of sediment-starved autoretreat. A sufficiently high sediment concentration can prevent sediment-starved autoretreat during the entire sea-level cycle. This observation may explain why some present-day river mouths are expressed in terms of deltas protruding into the sea, and others are wholly contained within embayments or estuaries in which water has invaded landward.