Yep.

"Goodbye, USA friend, I am emigrating to Tonga so I won't see you again."

"Hello my new Tongan neighbor! I am an immigrant to your beautiful country."

"Hey, buddies at my high school reunion, I'm doing great thanks! I am an expat living in Tonga these days."

"Emigrant" exists but almost unused (perhaps other than in specialist policy writing.) "Expat" is the common agent noun for "Emigrate."