In this paper, we apply mixture theory to quantitatively predict the transient behavior of drug delivery by using a microneedle array inserted into tissue. In the framework of mixture theory, biological tissue is treated as a multi-phase fluid saturated porous medium, where the mathematical behavior of the tissue is characterized by the conservation equations of multi-phase models. Drug delivery by microneedle array imposes additional requirements on the simulation procedures, including drug absorption by the blood capillaries and tissue cells, as well as a moving interface along its flowing pathway. The contribution of this paper is to combine mixture theory with the moving mesh methods in modeling the transient behavior of drug delivery into tissue. Numerical simulations are provided to obtain drug concentration distributions into tissues and capillaries.