In this paper we develop machinery for studying sequences of representations of any of the three families of classical Weyl groups, extending work of Church, Ellenberg, Farb, and Nagpal [7,8] on the symmetric groups Sn to the signed permutation groups Bn and the even-signed permutation groups Dn. For each family Wn, we present an algebraic framework where a sequence Vn of Wn-representations is encoded into a single object we call an FIW-module. We prove that if an FIW-module V satisfies a simple finite generation condition then the structure of the sequence is highly constrained. One consequence is that the sequence is uniformly representation stable in the sense of Church–Farb, that is, the pattern of irreducible representations in the decomposition of each Vn eventually stabilizes in a precise sense. Using the theory developed here we obtain new results about the cohomology of generalized flag varieties associated to the classical Weyl groups, and more generally the r-diagonal coinvariant algebras.We analyze the algebraic structure of the category of FIW-modules, and introduce restriction and induction operations that enable us to study interactions between the three families of groups. We use this theory to prove analogues of Murnaghan's 1938 stability theorem for Kronecker coefficients for the families Bn and Dn. The theory of FIW-modules gives a conceptual framework for stability results such as these.