Membranes with high selectivity offer an attractive route to molecular separations, where technologies such as distillation and chromatography are energy intensive. However, it remains challenging to fine tune the structure and porosity in membranes, particularly to separate molecules of similar size. Here, we report a process for producing composite membranes that comprise crystalline porous organic cage films fabricated by interfacial synthesis on a polyacrylonitrile support. These membranes exhibit ultrafast solvent permeance and high rejection of organic dyes with molecular weights over 600 g mol-1. The crystalline cage film is dynamic, and its pore aperture can be switched in methanol to generate larger pores that provide increased methanol permeance and higher molecular weight cut-offs (1,400 g mol-1). By varying the water/methanol ratio, the film can be switched between two phases that have different selectivities, such that a single, 'smart' crystalline membrane can perform graded molecular sieving. We exemplify this by separating three organic dyes in a single-stage, single-membrane process.