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One of the most striking successes of the lightcone bootstrap has been the perturbative computation of the anomalous dimensions and OPE coefficients of double-twist operators with large spin. It is expected that similar results for multiple-twist families can be obtained by extending the lightcone bootstrap to multipoint correlators. However, very little was known about multipoint lightcone blocks until now, in particular for OPE channels of comb topology. Here, we develop a systematic theory of lightcone blocks for arbitrary OPE channels based on the analysis of Casimir and vertex differential equations. Most of the novel technology is developed in the context of five- and six-point functions. Equipped with new expressions for lightcone blocks, we analyze crossing symmetry equations and compute OPE coefficients involving two double-twist operators that were not known before. In particular, for the first time, we are able to resolve a discrete dependence on tensor structures at large spin. The computation of anomalous dimensions for triple-twist families from six-point crossing equations will be addressed in a sequel to this work.