Does space-time 'flow' over time in the direction of its curvature?

I don't think most of the popular representations of curved spacetime are very good.

Space is flat (or a very good approximation to flat) - you only need to look out the window to see that. It is spacetime that is curved. If you want to visualise spacetime, you need to treat time in the same way as a spacial dimension. It helps if you are familiar with Minkowski diagrams.

In GTR straight lines are replaced with "geodesics", which look like straight lines "up close". (The earth's surface looks flat up close - it isn't) Objects move along geodesics (straight lines) like they normally do, but curvature gives the impression of gravity. I.e. gravity is a "fake" force.

Anyway you can "see" spacetime curvature in the spacetime diagram below. Time runs along the horizontal axis, height runs along the vertical axis. Ignoring the physical extension of the cannonball and friction etc...

• A cannonball is fired from a cannon at A. It moved along a geodesic (straight line in curved space) and drops to C.

• At the same time a cannon ball is dropped from A to B along another geodesic (straight line in curved space).

• When the cannon ball arrives at B, a 3rd cannon ball is fired at high speed along another geodesic (straight line in spacetime) and miraculously strikes the first cannonball at C.

The trajectory of the cannonballs forms a warped triangle whose interior angles sum to greater than 180 degrees, confirming that spacetime is curved in the presence of gravity.

The TicTok representation is inaccurate. In a spacetime, the earth would be a cylinder not a globe, with geodesics (e.g. trajectories of satellites) twisted around it. Note also that in GTR spacetime is "static" - everything has already "happened" in the same way everything has already happened in a movie. Einstein called time an illusion for this reason. So, spacetime does not "flow" whatever that means.

It is impossible to visualise an arbitrary curved 4D surface on a flat 2D surface like a computer screen. The diagram below is from the cover of Gravitation by Wheeler et. al. and probably the best diagram I've seen showing how curvature leads to gravitation. Hope that helps.

Do objects not just kove in the direction of spacetime curvature?

I'm not sure what you mean by "the direction of spacetime curvature". A direction is a vector whilst curvature is a tensor. The direction that a particle moves on a fixed spacetime is along a geodesic, the spacetime analogue of a straight line.