We report MINERvA’s first measurement of inclusive charged-current ${\bar{\nu }}_{\mu }$ cross sections on carbon, hydrocarbon, iron, and lead, and their ratios to the cross section on hydrocarbon, as functions of the antimuon transverse momentum, $p_T$. Using a wide-band ${\bar{\nu }}_{\mu }$ beam with mean energy $\sim 6\mathrm{\text{GeV}}$, these measurements probe all interaction modes, including the transition from resonance production to deep-inelastic scattering. The total uncertainties are typically $5$–$10\%$ for the absolute cross sections and $2$–$5\%$ for the ratios. Comparisons with multiple neutrino interaction models reveal significant discrepancies in the $p_T$ dependence, particularly for heavier nuclei. The disagreements are most pronounced at low $p_T$ but extend across the full $p_T$ range, indicating missing or mis-modelled nuclear effects.