We consider the cascade interconnection of two nonlinear time-delay systems, each of which being integral input-to-state stable (iISS). We provide an explicit growth condition on the dissipation rate of the driving system and the input rate of the driven system under which the overall cascade is itself iISS. Building upon recent iISS characterizations of time-delay systems, the method allows to consider Lyapunov-Krasovskii functionals (LKF) that dissipate in a point-wise manner along solutions, which simplifies its applicability as compared to dissipation rates involving the whole LKF itself. Another key feature of our approach is that the growth condition is imposed only on the driving state variables that actually appear in the LKF analysis of the driven subsystem: this feature happens to be new also in a delay-free context and helps reducing the conservatism of existing approaches. (C)& nbsp;2022 Elsevier Ltd. All rights reserved.