Horizontal gene transfer (HGT) is important for microbial evolution, but how evolutionary forces shape the frequencies of horizontally transferred genetic variants in the absence of strong selection remains an open question. In this study, we evolve laboratory populations of Acinetobacter baylyi (ADP1) with HGT from two clinically relevant strains of multidrug-resistant Acinetobacter baumannii (AB5075 and A9844). We find that DNA can cross the species barrier, even without strong selection, and despite substantial DNA sequence divergence between the two species. Our results confirm previous findings that HGT can drive the spread of antibiotic resistance genes (ARGs) without selection for that antibiotic, but not for all of the resistance genes present in the donor genome. We quantify the costs and benefits of horizontally transferred variants and use whole population sequencing to track the spread of ARGs from HGT donors into antibiotic-sensitive recipients. We find that even though most ARGs are taken up by populations of A. baylyi, the long-term fate of an individual gene depends both on its fitness cost and on the type of genetic element that carries the gene. Interestingly, we also found that an integron, but not its host plasmid, is able to spread in A. baylyi populations despite its strong deleterious effect. Altogether, our results show how HGT provides an evolutionary advantage to evolving populations by facilitating the spread of non-selected genetic variation including costly ARGs.