Here are two screen shots showing my first issue. In the first one you can see I've set a keyframe at frame 265 for the Nova parent group's position to be 0:0:
Then at frame 314 I set another keyframe changing this group's position to 200:0:
Except, at frame 314 (and all the frames in-between) the group is still visually at 0:0 instead of moving to the new position. There is a red dashed line showing the movement path, but the object doesn't visually move along that path. Even though the circle handle of the group's Position does move, the visual content of the group doesn't move. You can see the Nova replicator (colored black) is still centered in the content frame instead of shifted to the right.'
For the Slit Tunnel issue, I am changing the Center values under the Filters tab because it is the effect's center that I want to move. This does indeed change the effect's center, but when I do this then visually it also appears to be rotating because of the Caustics generator source. I think it's due to the how the generator's movement ends up interacting with changes to the Slit Tunnel's movement, but the "distortion" only occurs while the Slit Tunnel is moving - before and after the keyframes the tunnel looks "straight", but between the keyframes all the red "swirls" in the images above appear to rotate counter-clockwise even though no Rotation values are being changed. I could live with that, but it's not quite what I was going for with this.
The main thing to get working is the Nova group's position to change. Well, the Position does indeed change - the object's appearance in the frame doesn't change. No matter what I set its Position to, the Replicator still visually appears in the center of the frame.