  We revisit our study of general transport operator with general force field
and general invariant measure by considering, in the $L^1$ setting, the linear
transport operator $\T_H$ associated to a linear and positive boundary operator
$H$ of unit norm. It is known that in this case an extension of $\T_H$
generates a substochastic (i.e. positive contraction) $C_0$-semigroup
$(V_H(t))_{t\geq 0}$. We show here that $(V_H(t))_{t\geq 0}$ is the smallest
substochastic $C_0$-semigroup with the above mentioned property and provides a
representation of $(V_H(t))_{t \geq 0}$ as the sum of an expansion series
similar to Dyson-Phillips series. We develop an honesty theory for such
boundary perturbations that allows to consider the honesty of trajectories on
subintervals $J \subseteq [0,\infty)$. New necessary and sufficient conditions
for a trajectory to be honest are given in terms of the aforementioned series
expansion.
