Background: 6,7 Li and 9 Be are weakly bound against breakup into their cluster constituents. Breakup location is important for determining the role of breakup in above-barrier complete fusion suppression. Recent works have pointed out that experimental observables can be used to separate near-target and asymptotic breakup. Purpose: Our purpose is to distinguish near-target and asymptotic direct breakup of 6,7 Li in reactions with nuclei in different mass regions. Method: Charged particle coincidence measurements are carried out with pulsed 6,7 Li beams on 58 Ni and 64 Zn targets at sub-barrier energies and compared with previous measurements using 208 Pb and 209 Bi targets. A detector array providing a large angular coverage is used, along with time-of-flight information to give definitive particle identification of the direct breakup fragments. Results: In interactions of 6 Li with 58 Ni and 64 Zn, direct breakup occurs only asymptotically far away from the target. However, in interactions with 208 Pb and 209 Bi, near-target breakup occurs in addition to asymptotic breakup. Direct breakup of 7 Li into α-t is not observed in interactions with 58 Ni and 64 Zn. However, near-target dominated direct breakup was observed in measurements with 208 Pb and 209 Bi. A modified version of the Monte Carlo classical trajectory model code PLATYPUS, which explicitly takes into account lifetimes associated with unbound states, is used to simulate sub-barrier breakup reactions. Conclusions: Near-target breakup in interactions with 6,7 Li is an important mechanism only for the heavy targets 208 Pb and 209 Bi. There is insignificant near-target direct breakup of 6 Li and no direct breakup of 7 Li in reactions with 58 Ni and 64 Zn. Therefore, direct breakup is unlikely to suppress the above-barrier fusion cross section in reactions of 6,7 Li with 58 Ni and 64 Zn nuclei.