Although in relativity (where motion is relative and not absolute) we find that moving clocks run slow, this effect has nothing to do with the motion altering the way a clock works. With what does it have to do?

The first exposure to the relativity of simultaneity, the apparent shrinking of moving rods, and the slowing down of time may be disturbing. No observer’s measuring instruments or results are preferred over any others. Relativity gives us a wonderful symmetry between these observers; it does not assert the reality of slowing clocks, but rather that, from their two differing prospective, two observers in relative motion each observe that the other’s clock is slow. There is no necessity to grant preferred status to either of them, or to any other inertial observer.