Future of Bianchi I magnetic cosmologies with kinetic matter

We show under the assumption of small data that solutions to the Einstein-Vlasov system with a pure magnetic field and Bianchi I symmetry isotropise and tend to dust solutions. We also obtain the decay rates for the main variables. This generalises part of the work [V.~G.~LeBlanc, Classical Quantum Gravity 14, 2281-2301 (1997)] concerning the future behaviour of orthogonal perfect fluids with a linear equation of state in the presence of a magnetic field to the Vlasov case.