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Concern about gene flow from crops to wild relatives has become widespread with the increasing cultivation of transgenic crops. Possible consequences of such gene flow include genetic assimilation, wherein crop genes replace wild ones, and demographic swamping, wherein hybrids are less fertile than their wild parents, and wild populations shrink. Using mathematical models of a wild population recurrently receiving pollen from a genetically fixed crop, we find that the conditions for genetic assimilation are not stringent, and progress towards replacement can be fast, even for disfavoured crop genes. Demographic swamping and genetic drift relax the conditions for genetic assimilation and speed progress towards replacement. Genetic assimilation can involve thresholds and hysteresis, such that a small increase in immigration can lead to fixation of a disfavoured crop gene that had been maintained at a moderate frequency, even if the increase in immigration is cancelled before the gene fixes. Demographic swamping can give rise to 'migrational meltdown', such that a small increase in immigration can lead to not only fixation of a disfavoured crop gene but also drastic shrinkage of the wild population. These findings suggest that the spread of crop genes in wild populations should be monitored more closely.