We determine the relevance of Multi-Rate Mass Tansfer (MRMT) models to general diffusive porosity structures. To this end, we introduce Structured INteracting Continua models (SINC) as the combination of a finite number of diffusion-dominated interconnected immobile zones exchanging with an advection-dominated mobile domain. It directly extends Multiple INteracting Continua framework [34] by introducing a structure in the immobile domain, coming for example from the dead-ends of fracture clusters or poorly-connected dissolution patterns. We demonstrate that, whatever their structure, SINC models can be made equivalent in terms of concentration in the mobile zone to a unique Multi-Rate Mass Transfer (MRMT) model [21]. We develop effective shape-free numerical methods to identify its few dominant rates, that comply with any distribution of rates and porosities. We show that differences in terms of macrodispersion are not larger than 50% for approximate MRMT models with only one rate (double porosity models), and drop down to less than 0.1% for five rates MRMT models. Low-dimensional MRMT models accurately approach transport in structured diffusive porosities at intermediate and long times and only miss early responses.