Objects do not move towards regions of greater time dilation. In fact, the opposite is true: objects move along paths determined by the geometry of spacetime, which includes the effects of time dilation.
The motion of objects is influenced by the curvature of spacetime caused by mass and energy distributions. Massive objects curve the fabric of spacetime, creating what we perceive as gravitational forces. When an object is subject to a gravitational force, it follows the curved spacetime path dictated by the geometry.
The effect of time dilation is a consequence of this curved spacetime. In regions of stronger gravitational fields or regions with higher relative velocities, time runs slower compared to regions with weaker gravitational fields or lower relative velocities. This means that clocks closer to massive objects or clocks moving at higher speeds will tick more slowly compared to clocks further away from massive objects or moving at lower speeds.
It's important to note that objects do not actively move towards regions of greater time dilation. Rather, they follow the paths determined by the curvature of spacetime, and as a result, experience the effects of time dilation along those paths. The motion of objects is influenced by the geometry of spacetime, and time dilation is a consequence of that geometry.