Traditionally, censored quantile regression stipulates a specific, pointwise conditional quantile of the survival time given covariates. Despite its popularity owing to model flexibility and straightforward interpretation, the pointwise formulation oftentimes yields rather unstable estimates across neighbouring quantile levels with substantially large variances. In view of this phenomenon, we propose a new class of censored quantile regression models with time-dependent covariates that can capture the relationship between the failure time and the covariate processes of a target population that falls within a specific percentile bracket. The pooling of information within a homogeneous neighbourhood facilitates more stable, hence more efficient, estimates. Numerical studies demonstrate that the proposed estimator outperforms current alternatives under various settings in terms of smaller empirical bias and standard deviation. A perturbation-based resampling method is also provided to reconcile the asymptotic distribution of the parameter estimates. Finally, consistency and weak convergence of the proposed estimator are established via empirical process theory.