We present a porous medium model with applications to tissue movement and tumour growth. The model is based on the standard fluid mechanics approach to living tissues. We extend the analysis proposed in 2014 by Perthame, Quirós, and Vázquez, by incorporating the advective effects caused, for instance, by the presence of nutrients, oxygen, or a chemo-attractant. Passing to the singular limit for a stiff pressure law (incompressible limit), it is possible to connect a density- based model and a free-boundary problem of Hele-Shaw type. Our result extends known results due to weaker assumptions and a more general setting. In particular, we are able to recover the so- called complementarity relation, which allows to derive the pressure through an elliptic equation. To this end, we prove the strong compactness of the pressure gradient, blending two different techniques : an extension of the usual Aronson-Bénilan estimate in an L3-setting and an L4-uniform bound of the pressure gradient.