The need to assess agreement arises in many scenarios in biomedical sciences when measurements were taken by different methods on the same subjects. When the endpoints are survival outcomes, the study of agreement becomes more challenging given the special characteristics of time-to-event data. In this article, we propose a new framework for assessing agreement based on survival processes that can be viewed as a natural representation of time-to-event outcomes. Our new agreement measure is formulated as the chance-corrected concordance between survival processes. It provides a new perspective for studying the relationship between correlated survival outcomes and offers an appealing interpretation as the agreement between survival times on the absolute distance scale. We provide a multivariate extension of the proposed agreement measure for multiple methods. Furthermore, the new framework enables a natural extension to evaluate time-dependent agreement structure. We develop nonparametric estimation of the proposed new agreement measures. Our estimators are shown to be strongly consistent and asymptotically normal. We evaluate the performance of the proposed estimators through simulation studies and then illustrate the methods using a prostate cancer data example.