Comprehensive knowledge of mechanisms driving the acquisition of antimicrobial resistance is essential for the development of new drugs with minimized resistibility. To gain this knowledge, we combine experimental evolution in a continuous culturing device, the morbidostat, with whole genome sequencing of evolving cultures followed by characterization of drug-resistant isolates. Here, this approach was used to assess evolutionary dynamics of resistance acquisition against DNA gyrase/topoisomerase TriBE inhibitor GP6 in Escherichia coli and Acinetobacter baumannii. The evolution of GP6 resistance in both species was driven by a combination of two classes of mutational events: (i) amino acid substitutions near the ATP-binding site of the GyrB subunit of the DNA gyrase target; and (ii) various mutations and genomic rearrangements leading to upregulation of efflux pumps, species-specific (AcrAB/TolC in E. coli and AdeIJK in A. baumannii) and shared by both species (MdtK). A comparison with the experimental evolution of resistance to ciprofloxacin (CIP), previously performed using the same workflow and strains, revealed fundamental differences between these two distinct classes of compounds. Most notable were non-overlapping spectra of target mutations and distinct evolutionary trajectories that, in the case of GP6, were dominated by upregulation of efflux machinery prior to (or even in lieu) of target modification. Most of the efflux-driven GP6-resistant isolates of both species displayed a robust cross-resistance to CIP, while CIP-resistant clones showed no appreciable increase in GP6-resistance.