We study a growing biological tissue as an open biphasic mixture with mass exchange between phases. The solid phase is identified with the matrix of a porous medium, while the fluid phase is comprised of water, together with all the dissolved chemical substances coexisting in the pore space. We assume that chemical substances evolve according to transport mechanisms determined by kinematical and constitutive relations, and we propose to consider growth as a process able to influence transport by continuously varying the thermo-mechanic state of the tissue. By focusing on the case of anisotropic growth, we show that such an influence occurs through a continuous rearrangement of the tissue material symmetries. In order to illustrate this interaction, we restrict ourselves to diffusion-dominated transport, and we assume that the time-scales associated with growth and the transport process of interest are largely separated. This allows for performing an asymptotic analysis of the "field equations" of the system. In this framework, we provide a formal solution of the transport equation in terms of its associated Green's function, and we show how the macroscopic concentration of a given chemical substance is "modulated" by anisotropic growth.