[pfdietz]

The civilizations themselves don't have to be long lived, but once a star system is colonized it has to stay inhabited long enough. Civilizations could come and go, but not completely die out too quickly. As long as each colonized star system produces enough child colonies before it dies out the exponential growth will continue, with colonies eventually being reestablished at systems where former colonies may have expired.

Eventually there would be selection for colonies that can produce faster and/or longer lived colony ships, so they can plant new colonies faster. This will eventually shade over into the more conventional galactic colonization scenarios where stars can be treated as near stationary.