Unidirectional transport of liquids has attracted the attention of researchers in recent years for its wide application foreground. However, it is still a challenge to control the spreading of liquid, especially for oils with relatively high viscosity. In this paper, a flexible surface textured with branch-shaped microstructures is proposed. These asymmetric microstructures exhibit excellent unidirectional spreading behaviors for various oils. By suitably stretching the flexible surface to different stretch ratios, the spreading length of the oil droplets can be controlled. Moreover, the ongoing forward spreading of oil droplets can be suspended dynamically when the surface is stretched to 40%. Corresponding mechanism analysis demonstrates that surface stretching can narrow and close the microvalves between adjacent branches, which restrain the flow of the precursor film and the primary droplet. The switchable unidirectional spreading behavior enables the surface with such microstructures to be used for oil transportation, oil–water separation, and controllable lubrication.