Are 4-vectors covariant under general coordinate transformations?

In the context of special relativity, it is well known that 4-vectors are covariant under Lorentz transformations (which is a linear transformation in space-time), however are they covariant under general coordinate transformations (GCT)? Personally, I don’t see why they shouldn’t be. I mean in GR, when we refer to 4-vectors (or in general, tensors) we are talking about coordinate-free objects that are invariant under all coordinate transformations not just linear ones; so why the sudden emphasis on linear transformations in SR if the 4-vectors are invariant under GCT? Rarely do I see a textbook that touches upon non-Lorentzian transformations in SR, which strikes me as odd.

This also raises another question: If 4-vectors are indeed covariant under GCT then the relativistic Newton’s 2nd law $F=mA$ (where $F$ is the 4-force and $A$ is the 4-acceleration) should also be covariant under GCT, and in particular, valid in non-inertial frames given that it’s an equality between coordinate-free objects. Am i correct?