The following may be a worthwhile addition to the answer by contributor notovny.
The following simplified model is an instance of energy distributing in such a way that we can think of it as an even distribution.
Let's way we have a vessel filled with a diatomic gas. From outside we introduce heat. The molecules of the gas have a velocity distribution. Adding heat means the average of the velocity distribution goes up. The fact that the content is a diatomic gas makes a difference. There is the kinetic energy of spinning of the diatomic molecules. Some of the molecule-molecule collisinons are glancing collisions, and in such a collicion linear kinetic energy of motion in a straight line can transition to rotational kinetic energy of a molecule spinning.
Generally, the energy will tend to flow from high to low. So if a particular molecule happens to be spinning very fast, and it happens to nudge another slow molecule, then the most likely outcome is that the two molecules fly apart.
What tends to happen is that when a system has multiple degrees of freedom, then the energy content of that system tends to become redistributed in such a way that all degrees of freedom have the same level of energy.
Comparison: the case of communicating vessels. Fill any one of the set of communicating vessels, and over time the liquid becomed redistributed, such that across all vessels the level is equal.
Returning to energy:
An example on 1-to-1 distribution:
If you spin a liquid around the surface assumes a concave shape. At the perimeter the following two energies are larger than at the center:
-At the perimeter the kinetic energy is larger
-At the perimeter the liquid is higher, so the potential energy is larger
In fact, the outcome is that when the liquid is uniformly rotating then at every distance to the center of rotation the amount of kinetic energy (relative to the center of rotation) is equal to the amount of potential energy (relative to the center of rotation)
So there are cases of equipartition of energy, but the case of a projectile fired from a barrel is a different case altogether.