In the case of a moving point-like mass and in the linearized limit of a weak-gravity approximation these solutions of the Einstein equations are known as the Liénard–Wiechert gravitational potentials.
Wave-like solutions (variations) in gravitational field at any point of space at some instant of time t are generated by the mass taken in the preceding (or retarded) instant of time s < t on its world-line at a vertex of the null cone connecting the mass and the field point.
As in the case of the Liénard–Wiechert potentials for electromagnetic effects and waves, the static potentials from a moving gravitational mass (i.e., its simple gravitational field, also known as gravitostatic field) are "updated," so that they point to the mass's actual position at constant velocity, with no retardation effects.
Only gravitational waves, caused by acceleration of a mass, and which cannot be removed by a change in a distant observer's inertial frame, must be subject to aberration, and thus originate from a retarded position and direction, due to their finite velocity of travel from their source.
So long as no radiation is emitted, conservation of momentum requires that forces between objects (either electromagnetic or gravitational forces) point at objects' instantaneous and up-to-date positions, and not in the direction of their speed-of-light-delayed (retarded) positions.