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Continuous Time Markov Chains, Hawkes processes and many other interestingprocesses can be described as solution of stochastic differential equationsdriven by Poisson measures. Previous works, using the Stein's method, give theconvergence rate of a sequence of renormalized Poisson measures towards theBrownian motion in several distances, constructed on the model of theKantorovitch-Rubinstein (or Wasserstein-1) distance. We show that manyoperations (like time change, convolution) on continuous functions areLipschitz continuous to extend these quantified convergences to diffuse limitsof Markov processes and long-time behavior of Hawkes processes.

Diffusive limits of Lipschitz functionals of Poisson measures