It doesn't. A 4D object is a fixed figure in a 4D space. Just like a given triangle is a fixed figure on a 2D sheet of paper.
A path in 4D, and I'm assuming you're referring to Spacetime not a Euclidean space, is a world line. So it doesn't move it exists. Some authors do refer to things moving along a world line but I think this is for practicality.
Good point, cosmik debris. The object, continuum and "motion" if not given, will be presumed to be particles (or composites) from the Standard Model, 4D of spacetime (3+1) and velocity in distance/time respectively. Worldlines in a Minkowski spacetime diagram don't themselves move but they do indicate relative velocity as the inverse of slope in that reference frame.
One might construe the "movement" of a worldline but it would have to be defined as some kind of hyper-velocity*. I can imagine a use for such a construct in quantum mechanics, as it relates to phenomena where a single particle seems to pass through two slits at once. That is, to ascribe a wave nature to a particle.
*The term "hypervelocity" is however, already in use to indicate those > 3,000 m/sec.