A new numerical method called Vofire is described here. It can be used for integrating transport equations approximated on general meshes in any dimension. The algorithm relies on a piecewise constant reconstruc-tion of an unknown in each cell, where the reconstructed function varies only in the directions orthogonal to the velocity. This allows to design a non-dissipative and maximum preserving scheme for the transport of char-acteristic functions. It is then incorporated in a Lagrangian scheme with remeshing for the computation of multi-component compressible flows.