The analysis of the penalized Stokes problem, in its variable viscosity formulation, coupled to convection-diffusion equations is presented in this article. It models the interaction between a highly viscous fluid with variable viscosity and immersed moving and deformable obstacles. Indeed, while it is quite common to couple Poisson equations to diffusion-transport equations in plasma physics or fluid dynamics in vorticity formulations, the study of complex fluids requires to consider together the Stokes problem in complex moving geometry and convection-diffusion equations. The main result of this paper shows the existence and the uniqueness of the solution to this equations system with regularity estimates. Then we show that the solution to the penalized problem weakly converges toward the solution to the physical problem. Numerical simulations of fluid mechanics computations in this context are also presented in order to illustrate the practical aspects of such models: lung cells and their surrounding heterogeneous fluid, and porous media flows. Among the main original aspects in the present study, one can highlight the non linear dynamics induced by the coupling, and the tracking of the time-dependence of the domain.