Missing covariates often arise in biomedical studies with survival outcomes. Existing approaches for missing covariates generally assume proportional hazards. The proportionality assumption may not hold in practice, as illustrated by data from a mouse leukemia study with covariate effects changing over time. To tackle this restriction, we study the missing data problem under the varying-coefficient proportional hazards model. On the basis of the local partial likelihood approach, we develop inverse selection probability weighted estimators. We consider reweighting and augmentation techniques for possible improvement of efficiency and robustness. The proposed estimators are assessed via simulation studies and illustrated by application to the mouse leukemia data.