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We present a general class of covariate-adjusted response-adaptive (CARA) designs introduced in~\cite{Aletti.et.al.18}, which is based on a new functional urn model. We show strong consistency concerning the allocation probability and the proportion of subjects assigned to the treatment groups, in the whole study and for each covariate profile, allowing the distribution of the responses conditioned on covariates to be estimated nonparametrically. We also establish joint central limit theorems for these quantities and the joint sufficient statistics, which allow construction of inference procedures.