The continuous, constant-influx release of a dense fluid into a lighter one is investigated via a direct numerical simulation (DNS) study. The heavier fluid forms a gravity current that propagates along the horizontal bottom of a rectangular domain. The effect of the turbulent entrainment of the lighter ambient fluid into the gravity current on the mean flow dynamics is examined. The unsteady gravity current is horizontally divided into three regions with distinct entrainment characteristics: the hydraulic jump, the body, and the head. Classical entrainment hypotheses, originally developed for steady continuous currents, successfully model the entrainment mechanism of the unsteady continuous current along the hydraulic jumps. Downstream, along the body region and specifically in the wake of the head, the effects of billows shedding from the head preclude the applicability of the classical theory. Investigating the depth-averaged flow properties along the current reveals how its dynamics deviate from standard entrainment hypotheses. The relationship between the bulk entrainment coefficient across the hydraulic jump and the source properties is examined.