The importance of ancient long-distance migrations in shaping the geographical structure of genera and families is becoming ever more apparent. The long-distance immigrants were not random samples of their floras, but had attributes which made them prime candidates for the intercontinental sweepstakes. High propagule dispersability was one such trait. I propose that these invasive species also must have produced large numbers of propagules across their ranges, by virtue of large population numbers and sizes. They probably were widespread, major elements in their floras. These ideas are supported by the fact that propagule pressure is a prime determinant of a contemporary species' invasion potential, as is the size of its native geographical distribution. I propose that highly dispersable and propagule-rich lineages are likely to have high speciation rates, because access to new regions affords opportunities for ecological and geographical speciation. These lineages also may persist longer, being more broadly distributed in space. The evolutionary advantage of these lineages extends to periods of climatic change.