The present study provides the analysis of the generalized 3D Stokes problem in a time dependent domain. The fluid viscosity is nonlinear and depends on a transported and diffused quantity. This is a natural model of very low Reynolds number flow, typically at the microscale, involving a heterogeneous shear-thinning miscible medium transported and diffusing in a solvent filling a deformable geometry. This one-way coupling is meaningful when the action produced by the solid on the fluid is dominant or drives the flow. Several mathematical aspects are developed. The penalized version of this problem is introduced, involving the penalization of the solid in a deformable motion but defined in a simple geometry, which is of crucial interest for many numerical methods. All the equations of this partial differential system are analyzed separately, and then the coupled model is shown to be well-posed and to converge toward the solution of the initial problem. In order to illustrate the pertinence of such models, two meaningful micrometer scale real-life problems are presented: the dynamics of a polymer inside the pores of a real rock, in practice a Xanthan biodegradable polymer percolating a Bentheimer sandstone and miscible in water, and the dynamics of the strongly heterogeneous mucus bio-film, covering the human lungs surface, propelled by the vibrating ciliated cells. For these two examples, we show that the mathematical hypothesis are satisfied.