It has previously been shown that non-isothermal directional polymer crystallisation driven by local melting (Zone Annealing), has a close analogy with an equivalent isothermal crystallisation protocol. This surprising analogy is due to the low thermal conductivity of polymers—because they are poor thermal conductors, crystallisation occurs over a relatively narrow spatial domain while the thermal gradient spans a much wider scale. This separation of scales, which occurs in the limit of small sink velocity, allows replacing the crystallinity profile with a step and the temperature at the step acts as an effective isothermal crystallisation temperature. In this paper, we study directional polymer crystallisation under faster moving sinks using both numerical simulations and analytical theory. While, only partial crystallisation occurs, regardless, a steady state exists. At large velocity, the sink quickly moves ahead of a region that is still crystallizing; since polymers are poor thermal conductors, the latent heat dissipation to the sink becomes inefficient, eventually resulting in the temperature increasing back to the melting point thereby resulting in incomplete crystallization. This transition occurs when the two length scales measuring the sink-interface distance and the width of the crystallizing interface become comparable. For steady state and in the limit of large sink velocity, regular perturbation solutions of the differential equations governing heat transport and crystallization in the region between the heat sink and the solid–melt interface are in good agreement with numerical results.