The idea was first screen should have shown what you can do without needing a tutorial, but I did notice that sometimes the egg grab doesn't happen. But it does others, so... And I guess Doki just pulling herself to a bystander may not be as clear as I thought?
You have two basic things you can do: hookshot to the long-necked dragoons, allowing you to reach other places, or grab and lift to throw the chick and egg dragoons. the egg ones explode and make non-eggs passable (other eggs explode too)