We deal with the problem of uniform asymptotics in kernel functional estimation with the bandwidth depending on the data. In a unified approach, we investigate kernel estimates for the density and the hazard rate for uncensored and right-censored observations. The model allows for the fixed bandwidth as well as for various variable bandwidths, e.g., the nearest neighbour bandwidth. An elementary proof for the strong consistency of the general estimator is based on the local convergence of the empirical distribution function to the cumulative distribution function and the Nelson-Aalen estimator to the cumulative hazard rate. The result proves the strong consistency for yet untreated cases as, e.g., the hazard rate estimate with nearest neighbour bandwidth.