In the thought experiment to which you're referring, there is no general relativistic effects a priori. Rather, the idea is that you can use the fact that it is intuitive to derive the effect of acceleration of the elevator on wavelength of the light source to understand how acceleration can stretch or compress the wavelength of light.
This is an example of the Weak Equivalence Principle in action. Einstein says, if you want to understand how gravity effects light, you can first understand how acceleration effects light, then realize that acceleration should always be able to undo the effects of gravity, so the mathematical effects of accelerations and gravitational fields should be in one-to-one correspondence.
In practice, you can say something along the lines of "whenever you want to think about physics in a gravitational field, there is always an equivalent accelerating elevator problem you can consider."
Now if your elevator is travelling along a curved path, then you have the same instantaneous effects as you do in the straight line case.
To visualize, consider a scenario where the elevator had been at rest for some time, so that the peaks of the light wave can be seen as concentric spheres about the light source. Now consider what happens under an instantaneous acceleration. You change the picture by taking the sphere closest to the light source and "denting" it. The dent is inverse to the direction of the acceleration vector's projection onto the normal vector pointing out of the plane tangent to the sphere.
If you follow a curved path, you add up these effects for each instant. There are surely interesting patterns you can make, but fundamentally, you can still think about this problem as the sum of many linear instantaneous accelerations.