Background: $^{6,7}\mathrm{Li}$ and $^{9}\mathrm{Be}$ 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 $^{6,7}\mathrm{Li}$ in reactions with nuclei in different mass regions.Method: Charged particle coincidence measurements are carried out with pulsed $^{6,7}\mathrm{Li}$ beams on $^{58}\mathrm{Ni}$ and $^{64}\mathrm{Zn}$ targets at sub-barrier energies and compared with previous measurements using $^{208}\mathrm{Pb}$ and $^{209}\mathrm{Bi}$ 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 $^{6}\mathrm{Li}$ with $^{58}\mathrm{Ni}$ and $^{64}\mathrm{Zn}$, direct breakup occurs only asymptotically far away from the target. However, in interactions with $^{208}\mathrm{Pb}$ and $^{209}\mathrm{Bi}$, near-target breakup occurs in addition to asymptotic breakup. Direct breakup of $^{7}\mathrm{Li}$ into $\ensuremath{\alpha}\ensuremath{-}t$ is not observed in interactions with $^{58}\mathrm{Ni}$ and $^{64}\mathrm{Zn}$. However, near-target dominated direct breakup was observed in measurements with $^{208}\mathrm{Pb}$ and $^{209}\mathrm{Bi}$. 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 $^{6,7}\mathrm{Li}$ is an important mechanism only for the heavy targets $^{208}\mathrm{Pb}$ and $^{209}\mathrm{Bi}$. There is insignificant near-target direct breakup of $^{6}\mathrm{Li}$ and no direct breakup of $^{7}\mathrm{Li}$ in reactions with $^{58}\mathrm{Ni}$ and $^{64}\mathrm{Zn}$. Therefore, direct breakup is unlikely to suppress the above-barrier fusion cross section in reactions of $^{6,7}\mathrm{Li}$ with $^{58}\mathrm{Ni}$ and $^{64}\mathrm{Zn}$ nuclei.