A volcanic edifice generates a compressive stress field in the upper crust which affects magma transport. As a central cone builds up, primitive magmas can no longer erupt through the focal area and stall at depth. Within a vertical column terminating below the Earth's surface, magma is overpressured with respect to the surroundings and may feed a horizontally propagating dyke. We define three regimes for magma transport as a function of magma buoyancy, edifice size and density stratification in the upper crust: (1) eruption through the summit, (2) storage beneath the edifice, (3) horizontal propagation to feed a distal eruptive vent. We study the dynamics of horizontal dyke propagation away from the focal area of a volcanic field. Large horizontal propagation distances can only be achieved if an edifice prevents eruption through the focal area and if magma is negatively buoyant at shallow depth. With no edifice, a horizontal dyke is always tallest at the injection point, implying that, if it is able to breach the surface, it only does so in the focal area. With an edifice, at small distances from the axis, confining stresses due to the edifice load act to impede vertical propagation. In this case, a horizontal dyke does not grow vertically at the injection point and develops a hump at some distance from the focal area, which accounts for distal eruptive centers. During propagation, this hump may be far removed from the dyke tip and migrates at a different velocity. All else being equal, a decreasing supply rate or a decreasing magma viscosity generate eruptive centers at increasing distances from the focal area. This is consistent with the distribution of erupted products in many volcanic fields, such that, with increasing distance from the focal area, magma compositions are less and less evolved.