In this talk, I will build on the classification of EFTs with locally symmetric moduli spaces whose particle towers exhibit asymptotic exponential mass decay rates characteristic of either Kaluza–Klein modes or string oscillators, and which organise into one irrep of the duality group (see Mariana Graña's talk). I will show that these EFTs are related either through embeddings of one moduli space into another or via decompactification limits. Requiring compatibility of tower representations under the corresponding branching rules, we find that most theories can be obtained from an EFT based on (E_{8(8)}). However, three theories in the classification appear to be impossible to get from M- or string-theory compactifications. If time permits, I will also discuss how the same embedding procedure can be used to identify the string and brane representations expected to appear in the spectrum. Based on [2605.26212].