Fusion laser cutting allows for the processing of metallic sheets with high-edge quality, provided that process parameters are selected accurately. To guarantee quality while being robust to various existing uncertainties, velocity is typically set conservatively. This ensures complete cuts with limited defects such as low dross. However, such an approach significantly impacts productivity because the cutting velocity is empirically limited, often more than necessary. Literature has demonstrated that real-time dross estimation using the analysis of process emission images with Machine Learning algorithms, combined with control-based approaches, can effectively maximize productivity, while maintaining reference quality conditions. However, to date, this technique has been demonstrated only on linear cuts, limiting its industrial applicability. As a matter of fact, variations in the propagation of the process emission light within the coaxial monitoring chain, as well as intrinsic variations due to different cutting directions, significantly impact the performances of the estimation algorithm. This work presents an effective approach to extend the applicability of the velocity-based control strategy to multidirectional and curved geometries. A Neural Network was trained and tested to predict dross formation during linear cuts in different directions of 5 mm thick AISI304. The model predictions are robust, regardless of the direction of the cuts, with R2 values above 70% and limited Root Mean Square Error. The control architecture was then designed and tested on circular trajectories with variable curvatures, demonstrating resilient performance in terms of dross prediction and regulation. Finally, the controlled cut was tested on representative geometries, proving its industrial applicability.