From June's Dev Update:
Axie Core is about achieving connection between axies and their owners. Enabling this requires systems where your time, effort, energy, and skill each have an impact on your axie. Deeper vs. wider.
We estimate (with 75% confidence) that this process will start within the next few months through the release of axie experience points (AXP) and on-chain leveling. Axie OGs from 2018/2019 will be happy to learn that we’ll account for legacy experience points from the Auto-battle era within this system. AXP will be accumulated by playing with our axies, and it will be specific to the axies earning it (nontransferable).
We envision players participating in a wide variety of different Axie experiences in pursuit of AXP. Initially, we’ll experiment with AXP being made available through Origins. Then we’ll make it obtainable through other experiences.