I think the following is correct, but it's not from a textbook or paper, just my thoughts, and as I write this I realize I'm missing a few pieces to nail my ideas down. However, I'll write this anyway.
What we want to do is to see if there is any effect on the direction of spin of a gyroscope when there is a nearby mass current, due to a massive linear moving object flying nearby. This seems like the simplest sort of experiment which we might describe as "linear frame dragging".
Rather than do this directly, we can change the problem a bit, and make the massive object stationary and the gyroscope move by a suitable change of coordinates.
So, consider a gyroscope doing a flyby of a large mass. The gyroscope follows a geodesic, which is a curve that parallel transports itself. This curve has no proper acceleration, so there is no difference between parallel transport and Fermi-walker transport. Thus we conclude that if the gyroscope spin axis is pointed along the direction of motion initially (i.e. parallel to the direction of motion), it will remain parallel to the direction of motion. The gyroscope will parallel-transport and fermi-walker transport it's spin axis along the geodesic path of its orbit. This implies that the gyroscope spin axis rotates relative to the fixed stars since the direction of travel of the gyroscope changes during the flyby as its path is deflected. If we ignore various aberrations and other optical effects for the purpose of exposition , if we attach a telescope to the gyroscope pointed at a guide star, the telescope will not remain pointing at the guide star.
The result of this line of thought is that we conclude that the spin axis of the gyroscope rotates relative to the fixed stars. Also, we conclude that the angle of deflection depends on the orbit of the gyroscope, not how fast it spins or it's angular momentum. This does raise an interesting point, I suppose we'd really need to use the Papapetrou equations rather than the geodesic equations if the spin was high enough. I don't think it'd matter, but it might.
Now, consider a point of view of the gyroscope. Specifically, we consider the point of view of a local inertial frame attached to the gyroscope. (The idea of a point of view in GR is a bit ambiguous, so I am being more specific). The relative rotation of the spin axis to the fixed stars must be true in the gyroscope frame just as it was in the fame of the massive body. But we need a different explanation.
I am suggesting that this "something" that causes the gyroscopes spin access to change relative to the fixed stars is the gravitomagnetic field of the moving mass flying near the gyroscope, i.e. the moving mass generates a "mass current" which generates a gravitomagnetic field, which causes the spin access to change. But I haven't done a calculation to really justify this explanation by confirming that we can use it to numerically calculate the amount of deflection of the spin axis of the gyroscope relative to the fixed stars. Also, the analysis I used is rather idealized - for instance I am assuming the gyroscope is so small in extent that we can ignore effects like tidal torques which may be large in a realistic experimental flyby. And I've also assumed the spin isn't so large that it affects the travel path.
So, I haven't really done the necessary work to justify my intuitive conception, but I do think it's an interesting approach that might shed some light on the idea of "linear frame dragging".