Classical models of diffusion (thermal or chemical) typically treat flux and storage effects separately. This for example corresponds to Fickian diffusion. Nonlinear effects can be introduced by introducing a concentration/temperature dependence in the diffusion moduli, but this does not change the fundamental structure of the models. Recent multi-scale approaches considering transient effects at the micro-scale showed that quite different effective material response can emerge. Such effective response exhibit coupling between storage and transfer responses, and introduce history-dependence. In the linear case, semi-analytical models can be derived, either by enhanced mean-field approaches, or by modal representation. In the nonlinear case, the derivation of explicit models for the effective response remains an open issue. This difficulty in deriving explicit constitutive models precisely motivates data-driven approaches, and we will discuss in this contribution how this class of constitutive behaviours can be addressed in the framework of data-driven formulations. In the present case, the formulation should be written considering a constitutive space including not only gradients and fluxes, but also the primary field (concentration, entropy) and its dual (chemical potential, temperature). Nonlinearities and coupling between storage and transfer effects will then appear naturally. For more complex effective behaviour, exhibiting history dependence, the constitutive space must be extended, by rates and/or appropriate history variables. We will discuss different approaches, as sketched in earlier work.