  Probability currents are fundamental in characterizing the kinetics of
non-equilibrium processes. Notably, the steady-state current $J_{ss}$ for a
source-sink system can provide the exact mean-first-passage time (MFPT) for the
transition from source to sink. Because transient non-equilibrium behavior is
quantified in some modern path sampling approaches, such as the "weighted
ensemble" strategy, there is strong motivation to determine bounds on $J_{ss}$
-- and hence on the MFPT -- as the system evolves in time. Here we show that
$J_{ss}$ is bounded from above and below by the maximum and minimum,
respectively, of the current as a function of the spatial coordinate at any
time $t$ for one-dimensional systems undergoing over-damped Langevin (i.e.,
Smoluchowski) dynamics and for higher-dimensional Smoluchowski systems
satisfying certain assumptions when projected onto a single dimension. These
bounds become tighter with time, making them of potential practical utility in
a scheme for estimating $J_{ss}$ and the long-timescale kinetics of complex
systems. Conceptually, the bounds result from the fact that extrema of the
transient currents relax toward the steady-state current.
