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Proton beam dump experiments are among the most promising strategies to search for light and feebly interacting states such as axion-like particles (ALPs). The interpretation of these experiments is however complicated by the wide range of ALP models and the multitude of different production and decay channels that can induce observable signals. Here we propose a new approach to this problem by separating the calculation of constraints and projected sensitivities into model-independent and model-dependent parts. The former rely on extensive Monte Carlo simulations of ALP production and decays, as well as estimates of the detection efficiencies based on simplified detector geometries. Once these simulations have been performed and tabulated, the latter parts only require simple analytical rescalings that can be performed using the public code ALPINIST released together with this work. We illustrate this approach by considering several ALP models with couplings to Standard Model gauge bosons. For the case of ALPs coupled to gluons we show that the sensitivity of proton beam dump experiments can be extended significantly by considering hadronic ALP decays into three-body final states.