We perform a series of experiments to investigate the situation in which a melt-rich layer formed by a magma intrusion ascends through a crystalline magma chamber. The initial condition is such that a heavier granular layer overlies a liquid layer. The particles consisting the upper granular layer are in a jammed state, and only the particles near the interface can move to form a dilated boundary layer. The dilated layer detaches from the upper granular layer, and forms downwelling plumes which drive a cellular convection within the liquid-rich layer. The convection erodes the upper granular layer, and the liquid-rich layer migrates upwards with time. This upward migration of the liquid-rich layer differs from the previously known mechanisms of liquid transport; permeable flow in which the liquid migrates at the Darcy velocity, the Stokes settling in which the individual particle settles, and diapirs formed by the Rayleigh-Taylor instability. We find that the velocity of the upward migration of the liquid-rich layer can be scaled by the volumetric flux of the liquid ascending through the narrow channel between the particles. The upward migration of the liquid-rich layer is faster than the Darcy velocity. In a mushy magma chamber whose crystals are in a jammed state, neither the Stokes settling nor the Rayleigh-Taylor instability can occur. We propose that the upward migration of the melt-rich layer observed in our experiments can become an efficient mechanism of melt transport in a crystalline magma chamber.