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In this article, we tackle for the first time the problem of dynamic memory-efficient Searchable Symmetric Encryption (SSE). In the term "memory-efficient" SSE, we encompass both the goals of local SSE, and page-efficient SSE. The centerpiece of our approach is a novel connection between those two goals. We introduce a map, called the Generic Local Transform, which takes as input a page-efficient SSE scheme with certain special features, and outputs an SSE scheme with strong locality properties. We obtain several results. (1) First, for page-efficient SSE, we build a dynamic scheme with page efficiency $O(\log \log N)$ and storage efficiency $O(1)$, called LayeredSSE. The main technical innovation behind LayeredSSE is a new weighted extension of the two-choice allocation process, of independent interest. (2) Second, we introduce the Generic Local Transform, and combine it with LayeredSSE to build a dynamic SSE scheme with storage efficiency $O(1)$, locality $O(1)$, and read efficiency $O(\log\log N)$, under the condition that the longest list is of size $O(N^{1-1/\log \log λ})$. This matches, in every respect, the purely static construction of Asharov et al. presented at STOC 2016: dynamism comes at no extra cost. (3) Finally, by applying the Generic Local Transform to a variant of the Tethys scheme by Bossuat et al. from Crypto 2021, we build an unconditional static SSE with storage efficiency $O(1)$, locality $O(1)$, and read efficiency $O(\log^\varepsilon N)$, for an arbitrarily small constant $\varepsilon > 0$. To our knowledge, this is the construction that comes closest to the lower bound presented by Cash and Tessaro at Eurocrypt 2014.