  Non-Hermitian systems have garnered significant attention due to the
emergence of novel topology of complex spectra and skin modes. However,
investigating transport phenomena in such systems faces obstacles stemming from
the non-unitary nature of time evolution. Here, we establish the continuity
equation for a general non-Hermitian Hamiltonian in the Schr\"odinger picture.
It attributes the universal non-conservativity to the anti-commutation
relationship between particle number and non-Hermitian terms. Our work derives
a comprehensive current formula for non-Hermitian systems using Green's
function, applicable to both time-dependent and steady-state responses. To
demonstrate the validity of our approach, we calculate the local current in
models with one-dimensional and two-dimensional settings, incorporating
scattering potentials. The spatial distribution of local current highlights the
widespread non-Hermitian phenomena, including skin modes, non-reciprocal
quantum dots, and corner states. Our findings offer valuable insights for
advancing theoretical and experimental research in the transport of
non-Hermitian systems.
