You don't have to figure out a 1400-ly trip to be here. You just ("just") need to be able to move from one star system to the next.
If they took 100 years to get to the next star system 5-ly away and then 500 years to populate that system enough to start a new colonization, that's just ("just") 168,000 years to be here. Given a 2 billion year head start, that's easy.