[Enginerrrd]

Exactly. As soon as you magically remove the gravitational body, you are magically removing the waves too according to GR. There is no such thing as curved spacetime without mass. (Except for the cosmological constant, but that's different.)

[raattgift]

General Relativity admits general curved vacuum metrics (vacuum meaning: no matter anywhere), and many of them are useful theoretical approximations to real astrophysical systems. Famous ones include the Schwarzschild and Kerr metrics (both of which have T^{\mu\nu} = 0, where T is the stress-energy tensor), de Sitter and anti-de Sitter space, and Minkowski space. Useful ones include vacuum pp-waves, used in studying gravitational radiation from the perspective of an observer at large distance from the source. There's even the Sexl ultraboost, which can approximate ultrarelativistic motion between a black hole and a low-mass observer.

These are usually probed by adding test masses of some sort, letting them evolve along available trajectories. Some such test masses are pointlike, neutral, and nearly massless; others are some sort of classical or quantum field. In most cases, the goal is to keep T^{\mu\nu} negligible.

One can alternatively be lead by the stress-energy tensor, and may be tempted to call T^{\mu\nu} the matter tensor in that case. One typically chooses some vacuum background -- Minkowski space, usually, but any background can be used -- and then uses perturbation theory to capture how the chosen matter alters that background curvature. This is very common in cosmology.

> Except for the cosmological constant, but that's different

No, it's not different; one has flexibility to move the cosmological constant into the RHS for calculational convenience without having to change its interpretation as part of the background curvature: https://en.wikipedia.org/wiki/Lambdavacuum_solution