The population of North America is around 579 million after 500 years of exponential growth and probably the largest and longest sustained immigration trend in the worlds history with only open ocean as a barrier. We're talking about a star system with potentially multiple planets and moons let alone interplanetary space to colonize and century long gulfs of empty space to cross. My money is on it taking longer than 500 years before any native born son of a new star thinks to theirself that they could make a better life even further out.

I made the point in another comment but I do think there has to be a reason to send out colony ships. I agree that exploratory, unmanned probes would be sent out to gather information but I doubt sustained efforts to colonize new star systems would be made until there was significant social, political, or environmental pressure to do so. One ship colonies never succeed beyond a toe hold, you need a constant flow of people, resources and information to drive the founding of a new civilization especially if the trip is one way with little possibility of trade. If all you needed for the development of a new culture and society was an initial seed population and time then every town in the world should be a New York.