For \(S\in \Alg\slice R\) and \(M\in \mods{R}\), there is a base-change functor \(\Alg\slice R\to \Alg\slice S\) where \(X\mapsto X\tensor_R S\) that preserves many properties: e.g. if \(M\in \mods{R}^\fg\) then \(M\tensor_R S \in \mods{S}^\fg\). The reverse implication will hold if \(S\slice R\) is faithfully flat.
See descent data.