It seems to me that, as soon as there is mass in the universe which curves space-time, all paths of movement are closed. To a first approximation, for anything close to a single large mass that means an elliptical path.

That assumes, of course, that an instantaneous "path of movement" is something that is even worth considering, given that it ignores the time dimension and the fact that all those other objects aren't remaining in a static configuration around the object whose path we are considering.

I would think that parabolic trajectories only close in infinity, and hyperbolic not even there? Or both close only in infinity? The math on that is currently out of my brain's cache.

They all close. Worst case is they'll get longer and longer as the universe expands and the objects will never be able to complete them and come back for a second lap.