Generic 2D Hamiltonian systems possess partial barriers in their chaotic phase space that restrict classical transport. Quantum mechanically, the transport is suppressed if Planck's constant h is large compared to the classical flux, hΦ, such that wave packets and states are localized. In contrast, classical transport is mimicked for h Φ. Designing a quantum map with an isolated partial barrier of controllable flux Φ is the key to investigating the transition from this form of quantum localization to mimicking classical transport. It is observed that quantum transport follows a universal transition curve as a function of the expected scaling parameter Φ/h. We find this curve to be symmetric to Φ/h=1, having a width of 2 orders of magnitude in Φ/h, and exhibiting no quantized steps. We establish the relevance of local coupling, improving on previous random matrix models relying on global coupling. It turns out that a phenomenological 2×2 model gives an accurate analytical description of the transition curve.