Horizontal transfer of plasmids contributes substantially to the evolution of antimicrobial resistance in bacterial pathogens. In Staphylococcus aureus, most multiresistance plasmids (>20 kb) lack conjugation or mobilisation genes for horizontal transfer. However, most are indeed mobilisable due to carriage of origin-of-transfer (oriT) sequences that mimic those of conjugative plasmids related to pWBG749. pWBG749-family plasmids have diverged to carry distinct oriT subtypes and non-conjugative plasmids have also captured each of these subtypes, often in multiplicity. These oriT mimics are carried by 53% of non-conjugative S. aureus plasmids and therefore likely play a significant role in the spread of antimicrobial-resistance genes. Variants of the relaxasome accessory factor SmpO encoded by each conjugative plasmid subtype determine mobilisation specificity for these oriT subtypes. Conjugative plasmids that cannot normally mobilise a particular oriT subtype can be made to mobilise that subtype if the appropriate SmpO variant is present. In this work we characterised the oriT binding sites for five SmpO proteins using surface-plasmon-resonance (SPR)-based DNA-binding assays. SmpO proteins formed tetramers and hexamers in solution and bound to two 5’-GNNNNC-3’ sites spaced 65-67 basepairs apart within each oriT. Four of the five proteins bound their cognate sites only, revealing the molecular basis for mobilisation specificity. Phylogenetic comparisons suggested different lineages of pWBG749-family plasmids have switched oriT specificity between common oriT subtypes during their evolution. Mutagenesis revealed a single amino-acid position in the ribbon-helix-helix domain can switch oriT specificity in vivo and in vitro. Surprisingly, introduction of this change in pWBG749 reduced but did not abolish self-transfer and enabled mobilisation of a broader range of oriT subtypes. Thus, this variant may represent a promiscuous evolutionary intermediate in the DNA-specificity switch between oriT subtypes. We propose the apparent convergent evolution of oriT specificity in distinct branches of the pWBG749-family phylogeny reflects indirect selection pressure on pWBG749-family plasmids for a capacity to mobilise non-conjugative plasmids.