Volcanic eruptions transfer huge amounts of gas to the atmosphere. In particular, the sulfur released during large silicic explosive eruptions can induce global cooling. A fundamental goal in volcanology, therefore, is to assess the potential for eruption of the large volumes of crystal-poor, silicic magma that are stored at shallow depths in the crust, and to obtain theoretical bounds for the amount of volatiles that can be released during these eruptions. It is puzzling that highly evolved, crystal-poor silicic magmas are more likely to generate volcanic rocks than plutonic rocks. This observation suggests that such magmas are more prone to erupting than are their crystal-rich counterparts. Moreover, well studied examples of largely crystal-poor eruptions (for example, Katmai, Taupo and Minoan) often exhibit a release of sulfur that is 10 to 20 times higher than the amount of sulfur estimated to be stored in the melt. Here we argue that these two observations rest on how the magmatic volatile phase (MVP) behaves as it rises buoyantly in zoned magma reservoirs. By investigating the fluid dynamics that controls the transport of the MVP in crystal-rich and crystal-poor magmas, we show how the interplay between capillary stresses and the viscosity contrast between the MVP and the host melt results in a counterintuitive dynamics, whereby the MVP tends to migrate efficiently in crystal-rich parts of a magma reservoir and accumulate in crystal-poor regions. The accumulation of low-density bubbles of MVP in crystal-poor magmas has implications for the eruptive potential of such magmas, and is the likely source of the excess sulfur released during explosive eruptions.