  A family of heterogeneous mean-field systems with jumps is analyzed. These
systems are constructed as a Gibbs measure on block graphs. When the total
number of particles goes to infinity, a law of large numbers is shown to hold
in a multi-class context resulting in the weak convergence of the empirical
vector towards the solution of a McKean-Vlasov system of equations. We then
investigate the local stability of the limiting McKean-Vlasov system through
the construction of a local Lyapunov function. We first compute the limit of
adequately scaled relative entropy functions associated with the explicit
stationary distribution of the N-particles system. Using a Laplace principle
for empirical vectors we show that the limit takes an explicit form. Then we
demonstrate that this limit satisfies a descent property which, combined with
some mild assumptions shows that it is indeed a local Lyapunov function.
