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Abstract We present the first systematic study of strong binary–single and binary–binary black hole (BH) interactions with the inclusion of general relativity. By including general relativistic effects in the equations of motion during strong encounters, the dissipation of orbital energy from the emission of gravitational waves (GWs) can lead to captures and subsequent inspirals with appreciable eccentricities when entering the sensitive frequency ranges of the LIGO and Virgo GW detectors. It has been shown that binary–single interactions significantly contribute to the rate of eccentric mergers, but no studies have looked exclusively into the contribution from binary–binary interactions. To this end, we perform binary–binary and binary–single scattering experiments with general relativistic dynamics up through the 2.5 post-Newtonian order included, both in a controlled setting to gauge the importance of non-dissipative post-Newtonian terms and derive scaling relations for the cross section of GW captures, as well as experiments tuned to the strong interactions from state-of-the art globular cluster (GC) models to assess the relative importance of the binary–binary channel in facilitating GW captures and the resultant eccentricity distributions of inspiral from channel. Although binary–binary interactions are 10–100 times less frequent in GCs than binary–single interactions, their longer lifetime and more complex dynamics leads to a higher probability for GW captures to occur during the encounter. We find that binary–binary interactions contribute 25%–45% of the eccentric mergers that occur during strong BH encounters in GCs, regardless of the properties of the cluster environment. The inclusion of higher multiplicity encounters in dense star clusters therefore have major implications on the predicted rates of highly eccentric binaries potentially detectable by the LIGO/Virgo network. Because gravitational waveforms of eccentric inspirals are distinct from those generated by merging binaries that have circularized, measurements of eccentricity in such systems would highly constrain their formation scenario.