Liquid-liquid phase transition is occurring in many chemical engineering processes either as a desired phenomenon or as an undesired side effect. Typically, the phase split is modeled by neglecting all non-equilibrium effects. Here, a simple non-equilibrium situation is considered. Convective flow of a liquid along a decreasing temperature profile in a cooled channel is studied analytically. Three different scenarios for the transition process with two typical phase diagrams for binary mixtures are examined. For a phase diagram with critical concentration, phase segregation occurs via spinodal decomposition as a convective instability. For a cigar-shaped phase diagram the phase transformation is shown to evolve in analogy to directional solidification. Finger-like structures may be established under certain circumstances.