The opposite of stability, agilty is the ability to move from note to note. Having the freedom and flexibility to change notes over the range of the instrument. The greater the change, the more agility is needed. Imagine that same pyramid now turned up-side-down and balanced perfectly on its point. With the slightest effort, it could 'fall' into any direction. As brass players, we need to be able to change from note to note, and most of our demands include scaler (conjunct passages at a moderate tempo). As the repertoire gets more challenging, we will find greater leaps that happen at faster tempos. Developing agility is essential to navigate this repertoire.
As a musician (particularly a wind musician), we need an embouchure (when the muscles in our face create a stable formation that can allow a consistent flow of air to pass through an opening (aperture), which creates a consistent vibration in the lips, thus a consistent sound). The muscles in the face need to be coordinated in a very specific manner that allows for consistency as well as flexibility. This holds true for stability and agility.
As a brass musician, I cannot have too much agility. The greater my ability to move about the range of the instrument, the easier I will be able to navigate the repertoire. As long as my agility training doesn't start to effect my stability, agility can be viewed as almost limitless.
Most people will utilize lip slurs, lip trills, arpeggios, etc. to develop agility. These are all fine choices that will foster the development of agility. I also find it important to be familiar with contemporary tonalities, and therefore I utilize basic intervals (Major seconds, minor thirds, Major thirds, perfect fourths, etc.) up and down the instrument. This method requires solid understanding of theory- if you find it complicated, you need to do more theory homework before continuing.
Here is a video that further explains how I develop agility.