Multidimensional asymptotic normality of the kernel quantile estimator is established under fairly general conditions on the underlying distribution function and on the kernel. Sharpening these assumptions, one can utilize the proof to achieve also a bound for the rate of convergence which entails the comparison of the kernel estimator with the empirical quantile on the basis of their covering probabilities.
"Asymptotic Normality of the Kernel Quantile Estimator." Ann. Statist. 13 (1) 428 - 433, March, 1985. https://doi.org/10.1214/aos/1176346605