  We consider consistent particle systems, which include independent random
walkers, the symmetric exclusion and inclusion processes, as well as the dual
of the KMP model. Consistent systems are such that the distribution obtained by
first evolving $n$ particles and then removing a particle at random is the same
as the one given by a random removal of a particle at the initial time followed
by evolution of the remaining $n-1$ particles. In this paper we discuss two
main results. Firstly, we show that, for reversible systems, the property of
consistency is equivalent to self-duality, thus obtaining a novel probabilistic
interpretation of the self-duality property. Secondly, we show that consistent
particle systems satisfy a set of recursive equations. This recursions implies
that factorial moments of a system with $n$ particles are linked to those of a
system with $n-1$ particles, thus providing substantial information to study
the dynamics. In particular, for a consistent system with absorption, the
particle absorption probabilities satisfy universal recurrence relations. Since
particle systems with absorption are often dual to boundary-driven
non-equilibrium systems, the consistency property implies recurrence relations
for expectations of correlations in non-equilibrium steady states. We
illustrate these relations with several examples.
