Composting is a natural and forgiving process. About the only way you can stop it is to either add too much greens and/or water to the point that you lose the oxygen and it becomes anaerobic (and stinky), or if it dries out - at which point everything stops until water is supplied.

The only other outcome you can indirectly control is how fast it goes. (And again, regardless of speed, composting still happens as long as there is water and air - eventually. )

You start with browns. They will compost slowly on their own with a little water and air. You don't need anything else. Brown (high carbon) material like twigs, branches and leaves composts on its own on the forest floor all over the world and feeds forests, grasslands and meadows everywhere. It just takes 2 - 5 years to do it.

If you want to go faster (i.e. get hotter) you can add in some greens. Mix them with the browns and they can step on the gas and push the bacteria into high gear. The disadvantage of going faster is that you will lose some nutrients to the atmosphere when the bacteria are in hyperdrive. However, you can potentially cut your composting time in half, maybe more with an optimum C:N ratio.

But again - that is just optimizing for speed. You will actually get a higher quality product by going slow. (I always tell people that if you want the best soil amendment just put leaves in a pile and leave them for 2 - 3 years and you will have gardening gold - leaf mold - the amendment French gardeners have sworn by for hundreds of years.)

As for adding to the pile - you can add as much as you want whenever - it won't hurt or really affect what is already there. If you are trying to produce compost on a schedule, then remember that the compost pile won't truly be finished until the last thing you threw in there is done. Every time you dump something new in you are hitting the reset button as for the Time to Completion. If you want the thing to be done, at some point you have to close the lid, so to speak, and let it all process down.