You may have noticed that once you specify a crop in Lightroom and then try to move the crop frame, that it goes in the opposite direction from what you expect.  Instead of thinking of yourself clicking and dragging inside the crop frame  to move the crop frame, think of yourself as clicking and dragging inside the crop frame to move the photograph.   Once you adjust your thinking, you will find that it performs exactly as expected.

Define initial crop

Define initial crop

Clicked and dragged to the right to move photograph

Clicked and dragged to the right to move photograph