To reproduce:
1. Open attached project
2. Open scene "AvatarAssemblyTest"
3. Enter play mode - wait a bit until a character appears in scene. Note that he is in a T-pose
4. Select the character in hierarchy and in the inspector change any animator value - note that an idle animation starts playing. It should be playing by default (instead of the T-pose)

Explanation:
In this case you have to use DestroyImmediate(child.gameObject) because in the same frame you are parenting characterBody to characterBase and then bind the animator. This create a GO hierarchy that have both rig at the same time until the delay destroy call is process. Unfortunatly the animator try to bind the transform and pick the rig that is about to be deleted.