  We extend the construction by Kuelske and Iacobelli of metastates in
finite-state mean-field models in independent disorder to situations where the
local disorder terms are are a sample of an external ergodic Markov chain in
equilibrium. We show that for non-degenerate Markov chains, the structure of
the theorems is analogous to the case of i.i.d. variables when the limiting
weights in the metastate are expressed with the aid of a CLT for the occupation
time measure of the chain. As a new phenomenon we also show in a Potts example
that, for a degenerate non-reversible chain this CLT approximation is not
enough and the metastate can have less symmetry than the symmetry of the
interaction and a Gaussian approximation of disorder fluctuations would
suggest.
