I’m more inclined to say we move through the time dimension, though the question is sort of ill-defined if only because time, at least in theoretical physics, is more of a book keeping tool than anything else. It’s a bit hard to explain.
In any case, when we plot things on a grid and the axes are spatial like x, y, or z we’d say that it’s movement through space. We can easily plot motion on a time axis so I’d say that’s motion through time.
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