Through what? The invisible ropes connecting everything? The only thing that connects the Milky Way is gravity. Therefore, a large body from the galaxy first has to get close enough for its gravitational field to impact the Earth's orbit around the sun. That would be the "initial" change. If the edge of one galaxy starts going through the edge of the other side of the Milky Way, the Earth won't "feel" a thing.
There's also no such thing as "instant."






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