The central limit theorem is, with the strong law of large numbers, one of the two fundamental limit theorems in probability theory. Benjamin Jourdain and Alvin Tse have extended to non-linear functionals of the empirical measure of independent and identically distributed random vectors the central limit theorem which is well known for linear functionals. The main tool permitting this extension is the linear functional derivative, one of the notions of derivation on the Wasserstein space of probability measures that have recently been developed. The purpose of this work is to relax first the equal distribution assumption made by Jourdain and Tse and then the independence property to be able to deal with the successive values of an ergodic Markov chain.