The Green–Nagdhi equations are frequently used as a model of the wave-like behaviour of the free surface of a fluid, or the interface between two homogeneous fluids of differing densities. Here we show that their multilayer extension arises naturally from a framework based on the Euler–Poincaré theory under an ansatz of columnar motion. The framework also extends to the travelling wave solutions of the equations. We present numerical solutions of the travelling wave problem in a number of flow regimes. We find that the free surface and multilayer waves can exhibit intriguing differences compared to the results of single layer or rigid lid models.