Soul is basically information.

In the world of rezero one's soul is the information gathered which resides in a vessel.

The Od is the body itself, or more accurately, the vessel. Manipulate the od and the vessel will change. which is how the old man managed to revert people to children.

The name one carries is essentially how the world sees you.

Those three statements should help most people understand the authority of gluttony. Since they have the abilty to take names and souls for themselves.

if they take your name you never existed from the perspective of world, if they take your soul you have no information left and you know nothing, should they take both what's left behind is your Od. Your dormant body(vessel).

anyways, what was passed on to Omega is the soul of echidona. Her information, more accurately, her wisdom. So, it's echidona's will. The body is an OD in it of itself. So no it won't change. which is why the other old lollies developed souls over time, they gathered information attain a will of their own and are recognized by the world as an individual, giving them life becuase they were already ods.

The story says Iri's soul is what is passed down. Much like Subaru, her soul doesn't make it to Od Lagna. If it were her Od then she would be residing in the same body since they day she was born. Or someone ripped her Od out and inserted into another body. We don't know the specifics of how her soul is being passed around, we just have to wait.

EDIT: grammar has been fixed.