Thanks for the file, Nick.
Here is a file, which accesses the values given from the Time-Effector to the Object [the Cube in this case],
This information is easily available. If that data is fine with you, then have a look to the file below:
Scene file
https://www.amazon.com/clouddrive/share/OAunPavuEjbMnxE7HFbofbCUtZGMpg4uGO09QELK43x
But it doesn’t read the Cube’s axis, it tells what the Cube will receive. The Sample is taken from the Cube’s axis point (but not from “his” axis itself!) and that same value is given to the Null, as per your request [kind of]. It results in the same constellation. But it takes it from the source of animation, not inherited from the first receiver of it. I hope that makes sense.
Check the Coordinate manager, the cubes values do not change, while the Cube is rotating. So, it is not available, except for the CSTO [current state to object) procedure, and then frozen for one frame only. Again, no idea how to get this Cube change, for this set up.
The key is that the Effector is at the same location [PSR] as the cube, which brings up the idea to use the Global Matrix for the cube for the Sampling. This results in the feeling that the Global Matrix will change, based on the Time Effector. Now it becomes clear why the missing change can be taken as an advantage, as the Sample Matrix input is not changing. (Otherwise the effector would change the matrix and with this the results of the effoctor and the the cube would change twice, like a microphone/speaker feedback, not usable in other words.
Have a look in this scene
Scene file
https://www.amazon.com/clouddrive/share/Kh60zBmCodMvBP19pZfIGGSxXPRpUl7cgV977MXYSJV
The Time Effector uses as well a position change and the sphere is following accordingly, as the Tracer shows. Any previous value (Matrix) of the Cube will show up in the progress of the spheres [Tracer] path.
My best wishes