  Consider the global optimisation of a function $U$ defined on a finite set
$V$ endowed with an irreducible and reversible Markov generator.By integration,
we extend $U$ to the set $\mathcal{P}(V)$ of probability distributions on $V$
and we penalise it with a time-dependent generalised entropy
functional.Endowing $\mathcal{P}(V)$ with a Maas' Wasserstein-type Riemannian
structure, enables us to consider an associated time-inhomogeneous gradient
descent algorithm.There are several ways to interpret this $\cP(V)$-valued
dynamical system as the time-marginal laws of a time-inhomogeneous non-linear
Markov process taking values in $V$, each of them allowing for interacting
particle approximations.This procedure extends to the discrete framework the
continuous state space swarm algorithm approach of Bolte, Miclo and Villeneuve
\cite{Bolte}, but here we go further by considering more general generalised
entropy functionals for which functional inequalities can be proven.Thus in the
full generality of the above finite framework, we give conditions on the
underlying time dependence ensuring the convergence of the algorithm toward
laws supported by the set of global minima of $U$.Numerical simulations
illustrate that one has to be careful about the choice of the
time-inhomogeneous non-linear Markov process interpretation.
