I think there's some confusion here about what I mean. I will attempt to clarify.

The grasping reflex only occurs when something foreign touches a baby's hand or foot, not when the baby touches itself. It requires the processing resources of the brain to distinguish between the two, which is why this reflex is not mediated by the peripheral nervous system.

As the baby ages, the reflex - by which I mean the action of grasping - dwindles and ceases. It no longer occurs.

The parts of the brain responsible for triggering the action still exist and are known to continue to function. However, their final result no longer occurs.

When sufficient frontal lobe damage takes place, the reflex appears again, which suggests that they either counter the signal from the reflex center or override it.