The dispersal of non-native genes due to hybridization is a form of cryptic invasion with growing concern in evolution and conservation. This includes the spread of transgenic genes and antibiotic resistance. To investigate how genes and phenotypes are transmitted, we developed a general model that, for the first time, considers concurrently: multiple loci, quantitative and qualitative gene expression, assortative mating, dominance/recessivity inheritance and density-dependent demographic effects. Selection acting on alleles or genotypes can also be incorporated. Our results reveal that the conclusions about how hybridization threatens a species can be biased if they are based on single-gene models, while considering two or more genes can correct this bias. We also show that demography can amplify or balance the genetic effects, evidencing the need of jointly incorporating both processes. By implementing our model in a real case, we show that mallard ducks introduced in New Zealand benefit from hybridization to replace native grey-ducks. Total displacement can take a few generations and occurs by interspecific competition and by competition between hybrids and natives, demonstrating how hybridization may facilitate biological invasions. We argue that our general model represents a powerful tool for the study of a wide range of biological and societal questions.