In this paper, we study the local composite quantile regression estimator for mixed categorical and continuous data. The local composite quantile estimator is an efficient and safe alternative to the local polynomial method and has been well-studied for continuous covariates. Generalization of the local composite quantile regression estimator to a flexible data structure is appealing to practitioners as empirical studies often encounter categorical data. Furthermore, we study the theoretical properties of the cross-validated bandwidth selection for the local composite quantile estimator. Under mild conditions, we derive the rates of convergence of the cross-validated smoothing parameters to their optimal benchmark values for both categorical and continuous covariates. Monte Carlo experiments show that the proposed estimator may have large efficiency gains compared to the local linear estimator. Furthermore, we illustrate the robustness of the local composite quantile estimator using the Boston housing dataset.