The role of adaptation in determining invasion success has been acknowledged recently, notably through the accumulation of case studies of rapid evolution during bioinvasions. Despite this growing body of empirical evidence, there is still a need to develop the theoretical background of invasions with adaptation. Specifically, the impact of mating system on the dynamics of adaptation during invasion of a new environment remains only partially understood. Here, we analyze a simulation demo-genetic model of bioinvasion accounting for partial asexuality rates. We simulate two levels of recurrent immigration from a source population at mutation-drift-selection equilibrium to a new empty environment with a different adaptive landscape (black-hole sink). Adaptation relies on a quantitative trait coded explicitly by 10 loci under mutation, selection and genetic drift. Using this model, we confirm previous results on the positive effects on invasiveness of migration, mutation and similarity of local phenotypic optima. We further show how the invasion dynamics of the introduced population is affected by the rate of asexuality. Purely asexual species have lower invasion success in terms of probability and time to invasion than species with other mating systems. Among species with mixed mating systems, the greatest invasiveness is observed for species with high asexual rates. We suggest that this pattern is due to inflated genetic variance in the source population through the Hill-Robertson effect (i.e., clonal interference). An interesting consequence is that species with the highest genetic load in their source environment have greatest invasiveness in the new environment.