  Our knowledge on the active 3$\nu$ mixing angles ($\theta_{12}$,
$\theta_{13}$, and $\theta_{23}$) and the CP phase $\delta_{\mathrm{CP}}$ is
becoming accurate day-by-day enabling us to test the unitarity of the leptonic
mixing matrix with utmost precision. Future high-precision long-baseline
experiments are going to play an important role in this direction. In this
work, we study the impact of possible non-unitary neutrino mixing (NUNM) in the
context of next-generation long-baseline experiments DUNE and T2HKK/JD+KD
having one detector in Japan (T2HK/JD) and a second detector in Korea (KD). We
estimate the sensitivities of these setups to place direct, model-independent,
and competitive constraints on various NUNM parameters. We demonstrate the
possible correlations between the NUNM parameters, $\theta_{23}$, and
$\delta_{\mathrm{CP}}$. Our numerical results obtained using only far detector
data and supported by simple approximate analytical expressions of the
oscillation probabilities in matter, reveal that JD+KD has better sensitivities
for $|\alpha_{21}|$ and $\alpha_{22}$ as compared to DUNE, due to its larger
statistics in the appearance channel and less systematic uncertainties in the
disappearance channel, respectively. For $|\alpha_{31}|$, $|\alpha_{32}|$, and
$\alpha_{33}$, DUNE gives better constraints as compared to JD+KD, due to its
larger matter effect and wider neutrino energy spectrum. For $\alpha_{11}$,
both DUNE and JD+KD give similar bounds. We also show how much the bounds on
the NUNM parameters can be improved by combining the prospective data from DUNE
and JD+KD setups. We find that due to zero-distance effects, the near detectors
alone can also constrain $\alpha_{11}$, $|\alpha_{21}|$, and $\alpha_{22}$ in
both these setups. Finally, we observe that the $\nu_\tau$ appearance sample in
DUNE can improve the constraints on $|\alpha_{32}|$ and $\alpha_{33}$.
