In this paper we consider the modelling of nonlinear wave transformation by means of weakly nonlinear Boussinesq models. For a given couple linear dispersion relation-linear shoaling parameter, we show how to derive two systems of nonlinear PDEs differing in the form of the linear dispersive operators. In particular, within the same asymptotic accuracy, these operators can either be formulated by means of derivatives of the velocity, or in terms of derivatives of the flux. In the first case we speak of amplitude-velocity form of the model, in the second of amplitude-flux form. We show examples of these couples for several linear relations, including a new amplitude-flux variant of the model of Nwogu (J. Waterway, Port, Coast. Ocean Eng. 119, 1993). We then show, both analytically and by numerical nonlinear shoaling tests, that while for small amplitude waves the accuracy of the dispersion and shoaling relations is fundamental, when approaching breaking conditions it is only the amplitude-velocity or amplitude-flux form of the equations which determines the behaviour of the model, and in particular the shape and the height of the waves. In this regime we thus find only two types of behaviours, whatever the form of the linear dispersion relation and shoaling coefficient. This knowledge has tremendous importance when considering the use of these models in conjunction with some wave breaking detection and dissipation mechanism.Read less <