Estimating hazard ratios (HR) presents challenges for time-to-event analyses of cohorts with differential depletion of susceptibles. When the treatment effect is not null, cohorts that were balanced at baseline tend to become unbalanced on baseline characteristics over time as 'susceptible' individuals drop out of the population at risk differentially across treatment groups due to having outcome events. This imbalance in baseline covariates causes marginal (population-averaged) HRs to diverge from conditional (covariate-adjusted) HRs over time and systematically move toward the null. Methods that condition on a baseline propensity score (PS) yield HR estimates that fall between the marginal and conditional HRs when these diverge. Unconditional methods that match on the PS or weight by a function of the PS can estimate the marginal HR consistently but are prone to misinterpretation when the marginal HR diverges toward the null. Here, we present results from a series of simulations to help analysts gain insight on these issues. We propose a novel approach that uses time-dependent PSs to consistently estimate conditional HRs, regardless of whether susceptibles have been depleted differentially. Simulations show that adjustment for time-dependent PSs can adjust for covariate imbalances over time that are caused by depletion of susceptibles. Updating the PS is unnecessary when outcome incidence is so low that depletion of susceptibles is negligible. But if incidence is high, and covariates and treatment affect risk, then covariate imbalances arise as susceptibles are depleted, and PS-based methods can consistently estimate the conditional HR only if the PS is periodically updated.