In Dragon Age: The Veilguard ... the obvious choice here is to say that they should stay with the Grey Wardens. Their symbol is literally a Griffon (as seen above). After completing Davrins ...

The Grey Wardens ... him a Warden in Dragon Age 2, solidified my affection for them. The blue and white armor adorned with a mighty griffon emblem always felt like a symbol of hope and resilience.