The core-mantle boundary of Earth is a region where iron-rich liquids interact with oxides and silicates in the mantle. Iron enrichment may occur at the bottom of the mantle, leading to low seismic-wave velocities and high electrical conductivity, but plausible physical processes of iron enrichment have not been suggested. Diffusion-controlled iron enrichment is inefficient because it is too slow, although the diffusion can be fast enough along grain boundaries for some elements. More fundamentally, experimental studies and geophysical observations show that the core is under-saturated with oxygen, implying that the mantle next to the core should be depleted in FeO. Here we show that (Mg,Fe)O in contact with iron-rich liquids leads to a morphological instability, causing blobs of iron-rich liquid to penetrate the oxide. This morphological instability is generated by the chemical potential gradient between two materials when they are not in bulk chemical equilibrium, and should be a common process in Earth's interior. Iron-rich melt could be transported 50 to 100 kilometres away from the core-mantle boundary by this mechanism, providing an explanation for the iron-rich regions in the mantle.