Almost linear CFLGs can represent a substantial fragment of a Montague semantics for English and such "linear" grammar formalisms as (multi-component) tree-adjoining grammars (both as string grammars and as tree grammars) and multiple context-free grammars.
iO macro grammars and parallel multiple context-free grammars cannot be directly represented because representing string copying requires multiple occurrences of a variable of type o → o.