Having personally gone through a Pequot Naming Ceremony, and having seen personally Navajo, Iroquois, Zuni, Apache, Aztec, and Acoma naming ceremonies, I can tell you that a lot of the symbology behind the words is quite similar between the radically different groups.

Birds, in general, are symbols of peace, experience, and natural freedom. However, placing a bird, ANY BIRD, as "walking", indicates the opposite.

One could translate "Walking Eagle", to "One who breeds war and confinement".

By contrast, my name is "Soaring Hawk", translated as "One whose observations bring freedom".

I'm an inquisitive little bastard, and it has helped me shape my world view drastically.