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When you hover on the control points in the axis tool it turns yellow to indicate what is active, then click and drag.

Understanding the exact way that the object will move depends on the objects position, which is indicated by the axis tool's position. But you need to think in 3 dimensions.

In the 1st example I posted:  since the red & green axis are at an angle, the left/right movement would also be at the same angle (front to back). If the red was at an exact 180° (2nd picture), then the left/right movement would be straight across left or right, while the green would be back to front (90° to the red).

As the blue axis is at 90° in both examples the up/down movement would be straight up or down.

Maybe you knew all this already, but it might help out those who don't.