We give the first algorithms that, with high probability, maintain (1-ε)-approximate s-t maximum flow in an n-vertex undirected, capacitated graph undergoing either only edge insertions or only edge deletions in total update time Õ_ε(n²). For dense graphs, this yields polylogarithmic amortized update time, which was previously only obtained for the special case of uncapacitated graphs undergoing edge insertions. We develop the following two algorithms: - For graphs undergoing deletions, we generalize the congestion-balancing framework from [Aaron Bernstein et al., 2020], which was developed for maximum matching. We then show that this framework can be simulated on cut sparsifiers, which yields significant speed-ups. - For graphs undergoing insertions, we show that the sparsification techniques by Eppstein et al. [Eppstein et al., 1997] can be combined more directly with the techniques from Henzinger and Goranci [Goranci and Henzinger, 2023]. We thereby bypass the need to dynamize the more involved residual graph sparsification approach by Levin and Karger [Karger and Levine, 2015] suggested in [Goranci et al., 2025], and extend their result to capacitated graphs.