In this talk, we present the extension and application of the neural semi-Lagrangian methods developed in [1] to hyperbolic systems of balance laws. The core of our proposal lies in the integration of neural networks with kinetic relaxation techniques, an approximation that transforms the original nonlinear system into a larger set of decoupled linear transport equations, where nonlinearities are shifted to a source term and constant velocities satisfy the subcharacteristic condition. For the numerical resolution, we employ an operator splitting strategy. In the first stage, we solve the exact transport via the method of characteristics; in the second, we address the relaxation stage, where kinetic variables are projected toward local equilibrium, ensuring consistency with the macroscopic evolution of the original system. The central objective is to use neural networks to approximate n-parametric families of balance laws with parameters in the equations, initial, or boundary conditions, through a sequential training process. In this framework, at each time step, a network learns the evolution of the kinetic variables through transport and relaxation based on their state at the previous time. This semi-Lagrangian approach offers the competitive advantage of being CFL-less, allowing for larger time steps than traditional methods, and is implemented using first and second-order schemes (Strang splitting) optimized to reduce the number of trained networks without sacrificing precision. Furthermore, we discuss the adaptation of these schemes to be well-balanced [2] (preserving stationary solutions). To achieve high training precision, the Natural Gradient Descent (NGD) optimizer is utilized. Finally, we demonstrate the robustness of the method through a series of numerical tests applied to parametric families of the Burgers equations (with and without source terms), the shallow water equations (SWE) with bathymetry in 1D and 2D, and the Euler equations. [1] E. Franck, V. Michel-Dansac, L. Navoret, V. Vigon, Neural semi-Lagrangian method for highdimensional advection-diffusion problems, Computer Methods in Applied Mechanics and Engineering, 2026, 448 (B). [2] M. J. Castro, T. Morales de Luna, and C. Par´es, Well-balanced schemes and path-conservative numerical methods. In Handbook of Numerical Analysis 18, 1315-1341, Elsevier 2017.