In this talk I will present the main ideas of our proof of the stability of Minkowski spacetime for the massless Einstein-Vlasov system with non-compactly supported initial data. In particular, we will see how to differentiate the solutions in order to preserve the null structure of the equations and how to take advantage of it. I will also present certain hierarchies in the commuted equations and how to use them in order to deal with the slow decay rate of the metric. A similar result was proved by M. Taylor under a compact support assumption on the Vlasov field. This is joint work with David Fajman, Jérémie Joudioux, Jacques Smulevici and Maximilian Thaller.