"What happens to the additives in oil over time and is there any way to replenish them?"

Over time, additives are depleted performing the function for which they were intended, degraded by hydrolysis, mechanical shearing, condensation settling, water washing, particle scrubbing, etc.

The rate of depletion or degradation depends upon the application and the environment. In particular, heat, pressure, shear rate, fuel sulfur, soot, dirt, water, aeration and the presence of catalytic metals (copper, iron, etc.) affect the rate of depletion.

Regarding replenishment, whenever you top-up a system, you are replenishing additives. Likewise, you can perform a partial drain and replacement (often referred to as bleed-and-feed). 

A bleed-and-feed strategy can work if the base oil is not degraded. If the base oil has been degraded, adding new oil is analogous to sending a healthy person into a room full of sick people with the hope that his or her good health will be contagious. The additives in the new oil might be compromised within the first hours of use, leaving you right back where you started.

Casual addition of additives into a formulated oil can be dangerous and should be avoided. When in doubt, consult your lubricant supplier.