Path-following tasks, which refer to dynamic motion planning along pre-specified geometric references, are frequently encountered in applications such as milling, robot-supported measurements, and trajectory planning for autonomous vehicles. Different convex and non-convex optimal control formulations have been proposed to tackle these problems for the case of perfect models. This paper analyzes path-following problems in the presence of plant-model mismatch. The proposed adaptation strategies rely on concepts that are well known in the field of real-time optimization. We present conditions guaranteeing that, upon convergence, a minimum-time solution is attained despite the presence of plant-model mismatch. We draw upon a simulated robotic example to illustrate our results.