Automating drone-assisted processes is a complex task. Many solutions rely on trajectory generation and tracking, whereas in contrast, path-following control is a particularly promising approach, offering an intuitive and natural approach to automate tasks for drones and other vehicles. While different solutions to the path-following problem have been proposed, most of them lack the capability to explicitly handle state and input constraints, are formulated in a conservative two-stage approach, or are only applicable to linear systems. To address these challenges, the paper is built upon a Model Predictive Control-based path-following framework and extends its application to the Crazyflie quadrotor, which is investigated in hardware experiments. A cascaded control structure including an underlying attitude controller is included in the Model Predictive Path-Following Control formulation to meet the challenging real-time demands of quadrotor control. The effectiveness of the proposed method is demonstrated through real-world experiments, representing, to the best of the authors' knowledge, a novel application of this MPC-based path-following approach to the quadrotor. Additionally, as an extension to the original method, to allow for deviations of the path in cases where the precise following of the path might be overly restrictive, a corridor path-following approach is presented.