Do gravitational waves travel always in a straight line (along a geodesic) like EM waves? There are a lot of questions and answers on this site about light traveling in straight lines in vacuum (following a geodesic). And there are a lot about both EM and gravitational waves traveling at the same speed $c$.
I have read this question:

When we look at light propagating in the classical limit then it travels in straight lines.

How do single photons travel from here to there
And this one:

the gravitational wave paths are the same as light paths

Do gravitational lenses work on gravitational waves?
Now based on these, gravitational waves should always travel in a straight path (follow geodesics), just like EM waves. Actually, this is what we call a null geodesic.
Why is light described by a null geodesic?
But is this correct, that gravitational waves must travel in straight lines always, following null geodesics, and can this be proven?
As per the comment, the question is more interesting, because geodesics follow the curvature of spacetime, and gravitational waves are perturbations of spacetime itself.
Question:

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*Do gravitational waves travel always in a straight line (along a geodesic) like EM waves?

 A: In this link the similarities and differences of gravitational waves to electromagnetic waves is explored .
You ask:

Do gravitational waves travel always in a straight line like EM waves?

I think that the straight lines are the light ray description of light ,

In optics a ray is an idealized model of light, obtained by choosing a line that is perpendicular to the wavefronts of the actual light

There are equivalent rays defined for gravitational waves and that is the geometric optics, which  when wavelengths become too large for the system studied have to be modified, as this link suggests.

It is standard practice to study the lensing of gravitational waves (GW) using the geometric optics regime. However, in many astrophysical configurations this regime breaks down as the wavelength becomes comparable to the Schwarzschild radius of the lens.

There is also this link:

The geometrical-optics expansion reduces the problem of solving wave equations to one of the solving transport equations along rays. Here, we consider scalar, electromagnetic and gravitational waves propagating on a curved spacetime in general relativity. We show that each is governed by a wave equation with the same principal part. It follows that: each wave propagates at the speed of light along rays (null generators of hypersurfaces of constant phase) ......

A: 
the gravitational wave paths are the same as light paths

I cannot agree with this quote.
Light or more generally EM radiation consists of photons. These quanta do not dissipate from their beginning (emission) to their end (absorption). A photon beam from a laser, for example, is limited in diameter and the focus is not so perfect that this diameter will remain constant.
The diameter of the light beam increases over distance, but the gravitational potential as a kind of medium has (almost) nothing to do with it. The energy content of the quanta does not change, the number of quanta does not change and they do not dissipate along their geodesic path.
The same cannot be said about the gravitational potential. Whether expressed by gravitons or not, the gravitational potential above the scale of gravitons is a continuum. A mountain on a celestial body will never result in a discontinuity for a spacecraft orbiting such a body. It follows that gravity is dissipative in space.
