We develop a new algorithmic framework for designing approximation algorithms for cut-based optimization problems on capacitated undirected graphs that undergo edge insertions and deletions. Specifically, our framework dynamically maintains a variant of the hierarchical $j$-tree decomposition of [Madry FOCS'10], achieving a poly-logarithmic approximation factor to the graph's cut structure and supporting edge updates in $O(n^ε)$ amortized update time, for any arbitrarily small constant $ε\in (0,1)$. Consequently, we obtain new trade-offs between approximation and update/query time for fundamental cut-based optimization problems in the fully dynamic setting, including all-pairs minimum cuts, sparsest cut, multi-way cut, and multi-cut. For the last three problems, these trade-offs give the first fully-dynamic algorithms achieving poly-logarithmic approximation in sub-linear time per operation. The main technical ingredient behind our dynamic hierarchy is a dynamic cut-sparsifier algorithm that can handle vertex splits with low recourse. This is achieved by white-boxing the dynamic cut sparsifier construction of [Abraham et al. FOCS'16], based on forest packing, together with new structural insights about the maintenance of these forests under vertex splits. Given the versatility of cut sparsification in both the static and dynamic graph algorithms literature, we believe this construction may be of independent interest.