A new kind of underwater vehicle is developed by taking inspiration from cephalopods. Its actuation routine is scrutinized via a suitable model. Similar to octopuses and squids, these vehicles consist of an elastic, hollow shell capable of undergoing sequential stages of ingestion and ejection of ambient fluid, which is driven by the recursive inflation and deflation of the shell. The shell actively collapses, and in this way it expels water through a funnel; then it passively returns to the inflated shape, drawing ambient fluid into the cavity. By doing so, a pulsed-jet propulsion routine is performed that enables the vehicle to propel itself in water. Due to their soft nature, the actuation of these vehicles is largely dependent on the subtle management of the elastic response of the shell throughout the propulsion routine. A kinematic model of the actuation mechanism, thoroughly corroborated by experimental validation, is devised which elucidates the relationship between the active (collapse) and passive (refill) stages of the actuation. Upon association with the dynamics of the vehicle, this model permits the derivation of the generic performance profiles of this new kind of vehicle. It is acknowledged that, for given design specifications, an optimal swimming speed exists in coincidence with the coordinated operation between the crank mechanism driving the shell contraction and the onset of elastic energy, which determines the speed of inflation of the shell. These results are invaluable in the definition of rigorous design criteria and derivation of ad-hoc control laws for a new breed of optimized soft-bodied, pulsed-jet, unmanned underwater vehicles.