Nature-inspired flapping foils have attracted interest for their high thrust efficiency, but the large motions of their boundaries need to be considered. It is challenging to develop robust, efficient grid deformation algorithms appropriate for the large motions in three dimensions. In this paper, a volume grid deformation code is developed based on finite macro-element and transfinite interpolation, which successfully interfaces to a structured multi-block Navier-Stokes code. A suitable condition that generates the macro-elements with efficiency and improves the robustness of grid regularity is presented as well. As demonstrated by an airfoil with various motions related to flapping, the numerical results of aerodynamic forces by the developed method are shown to be in good agreement with those of an experimental data or a previous numerical solution.