Many Eidolons cannot actually pass into or directly influence the Material plane of Azuria . There is a ritualistic means through which they can overcome this rule, but it requires a physical sacrifice from a mortal native to the Material plane.
This sacrifice is often something simple like a hand, or an eye, but can be at times as violent as a heart, a soul, or even a brain. One the Eidolon possesses such a trinket, they may fuse it to their own essence and enter into the Material plane directly.
This ritual works for all planes, and is not limited to the material plane.
If an Eidolon is slain within the plane, they are once again banished from it until they gain another trinket. It is said that the only way to truly kill an Eidolon is to systematically hunt it down throughout all the major planes of existence, and slay it there until it has no planes to which it may enter. If this has ever come to pass, it is uncertain.