The translational-rotational motion of a triaxial body with constant dynamic shape and variable size and mass in a non-stationary Newtonian central gravitational field is investigated. Differential equations of motion of the triaxial body in the relative coordinate system with the origin at the center of a non-stationary spherical body are obtained. The axes of the Cartesian coordinate system fixed to the non-stationary triaxial body are coincident with its principal axes and their relative orientation is assumed to remain unchanged in the course of evolution. An analytical expression for the force function of the Newtonian interaction of the triaxial body of variable mass and size with a spherical body of variable size and mass is obtained. Differential equations of translational-rotational motion of the non-stationary triaxial body are derived in Jacobi osculating variables and are studied with the perturbation theory methods. The perturbing function is expanded in power series in terms of the Delaunay--Andoyer elements up to the second harmonic element inclusive. The evolution equations of the translational-rotational motion of the non-stationary triaxial body are obtained in the osculating elements of Delaunay--Andoyer.