Various smoothing methods for quantile density estimation are unified into a generalized kernel smoothing. Based on a stochastic upper bound of the derivatives sequence for a sequence of smoothed Brownian bridges, uniform in-probability consistency of generalized kernel quantile density estimators on any closed subinterval of the open unit interval is derived.
"Uniform consistency of generalized kernel estimators of quantile density." Ann. Statist. 23 (6) 2285 - 2291, December 1995. https://doi.org/10.1214/aos/1034713657