A field-theoretic model describing nonrelativistic four-body scattering processes is developed. The model is related to Bronzan's extended Lee model but the allowed interactions are restricted so that the resulting dynamical equations are as simple as possible, yet still exact. Two elementary particles n and a are introduced with the couplings n+n↔D and a+a↔C. Three-particle processes are generated by the additional coupling D+a↔α, leading to the possible three-body reactions D+a→D+a and D+a→n+n+a. The four-body sector then involves the 2→2 reactions aα→aα and aα→CD, the 2→3 reactions aα→Daa and aα→Cnn, and the 2→4 reaction aα→nnaa. Off-shell integral equations are obtained for the 2→2 amplitudes, and from these, expressions for the 2→3 and 2→4 amplitudes are constructed. Possible applications and generalizations of the model are discussed.