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We consider the problem of predictive modeling from irregularly and sparsely sampled longitudinal data with unknown, complex correlation structures and abrupt discontinuities. To address these challenges, we introduce a novel inducing clusters longitudinal deep kernel Gaussian Process (ICDKGP). ICDKGP approximates the data generating process by a zero-mean GP with a longitudinal deep kernel that models the unknown complex correlation structure in the data and a deterministic non-zero mean function to model the abrupt discontinuities. To improve the scalability and interpretability of ICDKGP, we introduce inducing clusters corresponding to centers of clusters in the training data. We formulate the training of ICDKGP as a constrained optimization problem and derive its evidence lower bound. We introduce a novel relaxation of the resulting problem which under rather mild assumptions yields a solution with error bounded relative to the original problem. We describe the results of extensive experiments demonstrating that ICDKGP substantially outperforms the state-of-the-art longitudinal methods on data with both smoothly and non-smoothly varying outcomes.