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 new 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. - First, we build a dynamic SSE scheme with storage efficiency O (1) and page efficiency only O (log log(N/p)), where p is the page size, called LayeredSSE. The main technique behind LayeredSSE is a new weighted extension of the two-choice allocation process, of independent interest. - 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. from STOC 2016: dynamism comes at no extra cost. - 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 ε N), for an arbitrarily small constant ε > 0. To our knowledge, this is the construction that comes closest to the lower bound presented by Cash and Tessaro at Eurocrypt 2014.