The dynamics of magma flow through a reservoir connected to a fissure in an elastic medium is investigated. Magma is supplied to the reservoir at a rate which may vary with time. The distribution of fluid pressure, the elastic deformation of the reservoir walls and the characteristics of dyke intrusion are calculated for a range of reservoir sizes and supply rates. Magma is stored in the reservoir whilst magma is intruded in the fissure. At high supply rates, the reservoir pressure increases by large amounts and only small amounts of magma flow into the fissure. Conversely, at low supply rates, most of the magma supplied to the system is injected into the fissure. For a given supply rate, the amount stored depends weakly on the reservoir size, and is largest for small reservoirs. The delay between the onset of reservoir inflation (i.e., the start of magma input) and the opening of the fissure decreases with increasing reservoir size. Rapid deflation of the reservoir occurs if the supply rate decreases with time. All else being equal, the largest reservoir pressures are reached for high supply rates, which suggests that these conditions lead to summit eruptions. Rift-zone intrusions are most extensive at low supply rates.