Aren't all processes stable? Well no, they're not and there's no escaping entropy, that natural tendency for disorder to increase. Unless they're mission critical, processes are often thought to be set and forget which is why they're always needing constant improvement. When you've gone to the trouble of designing a new process or improving an existing one, don't take a holiday until the controls are in place to keep it performing as specified.
At a minimum, key measures should be defined and a measurement plan enacted, both for the process itself and any associated processes that you know might be affected. Remember, no process is an island. Excel charts may look impressive, but try to get some time series Control Charts in place, as these are more sensitive to variation in performance, and analysis of their patterns can provide you with some insight into the causes of any variation that might be occurring. They are based on sound statistical principles and who doesn't love statistics? OK, there's no need to answer that …