We have studied the motion of liquid droplets located in the contact zone of two soft solids that are pressed together inside a liquid. The driving force of the motion is the gradient of elastic contact pressure. Upon reaching the perimeter of the contact spot, the droplets coalesce with the surrounding bulk liquid either in a continuous or discontinuous fashion. Droplets that exhibit complete coalescence move with a speed proportional to the pressure gradient along their trajectory. However, droplets that undergo partial coalescence cascades (up to 9 were observed) do not. The latter also move significantly slower. We have identified surface roughness as a potential mechanism causing the contrasting behavior.