The present work aims to introduce a method of unstructured moving grid generation with an emphasis on the capability of simulating moving body flow problems and being produced automatically at each time step according to the position of the body, which arises in many engineering fields involving multiphase flows, including sedimentation, improvement of combustion, or reducing the erosion by small droplets in large turbines. The proposed technique is based on the finite element method. The generated grid was applied in the case of moving a tetrahedron body in the computational domain. In this regard, the information of the grid was mapped from each time step to the next time step with the help of an algorithm of data interpolation. Additionally, to reduce the calculations, a two-block grid was generated in which total remeshing took place only in the areas close to the moving body, and in more remote areas, grid regeneration was performed only in the required areas.