The lattice Boltzmann equation (LBE), rooted in kinetic theory, provides a powerful framework for capturing complex flow behaviour through the evolution of single-particle distribution functions (PDFs). Despite its success, numerical solution of the LBE remains computationally demanding due to the small time steps imposed by explicit kinetic schemes and CFL type stability constraints. In this work, we introduce a physics-informed neural operator framework for the LBE that enables accurate prediction over large time horizons without step-by-step time integration, effectively bypassing the explicit evaluation of the collision kernel. The approach incorporates intrinsic moment-matching constraints of the LBE together with global equivariance properties of the full distribution field, allowing the model to capture the essential dynamics of the underlying kinetic system. The proposed framework is discretization-invariant, permitting models trained on coarse-grained PDFs to generalize to finer spatial resolutions, even when the relaxation-time parameter differs between coarse and fine lattices. Moreover, the method is agnostic to the specific lattice Boltzmann formulation, enabling the same neural operator architecture and loss design to be applied across distinct kinetic datasets governed by different dynamics. Our results demonstrate robustness across a range of challenging flow scenarios, including von Karman vortex shedding, ligament breakup, and bubble adhesion, establishing a data-driven pathway for fast and accurate modelling of kinetic systems.