Learning guarantees often rely on assumptions of i.i.d. data, which will likely be violated in practice once predictors are deployed to perform real-world tasks. Domain adaptation approaches thus appeared as a useful framework yielding extra flexibility in that distinct train and test data distributions are supported, provided that other assumptions are satisfied such as covariate shift, which expects the conditional distributions over labels to be independent of the underlying data distribution. Several approaches were introduced in order to induce generalization across varying train and test data sources, and those often rely on the general idea of domain-invariance, in such a way that the data-generating distributions are to be disregarded by the prediction model. In this contribution, we tackle the problem of generalizing across data sources by approaching it from the opposite direction: we consider a conditional modeling approach in which predictions, in addition to being dependent of the input data, use information relative to the underlying data-generating distribution. For instance, the model has an explicit mechanism to adapt to changing environments and/or new data sources. We argue that such an approach is more general than current domain adaptation methods since it does not require extra assumptions such as covariate shift and further yields simpler training algorithms that avoid a common source of training instabilities caused by minimax formulations, often employed in domain-invariant methods.