Swami Desikan had dealt about poorna, avesha and shakti incarnations in Tatvatrayam.
Emperuman himself appeared as Rama and Krishna. These are poorna avatarams
Many of his incarnations are Avesha Avataram; he pervades into a Jeevan or infuses his power (shakti) into a jeevan to accomplish His divine will. Perumal selects such beings.
For example, Parasurama still exists; but his incarnation had ended. That is, Perumal had infused his strength onto him to fulfill the divine enactments and so are Vyasa and others. One can know about this elaborately in Srimad Rahasyatrayasaram.