Experimentally detected ultrafast spin-avalanches spreading in crystals of molecular (nano)magnets (Decelle et al., Phys. Rev. Lett. 102, 027203 (2009)), have been recently explained in terms of magnetic detonation (Modestov et al., Phys. Rev. Lett. 107, 207208 (2011)). Here magnetic detonation structure is investigated by taking into account transport processes of the crystals such as thermal conduction and volume viscosity. In contrast to the previously suggested model, the transport processes result in smooth profiles of the most important thermodynamical crystal parameters - such as temperature, density and pressure - all over the magnetic detonation front including the leading shock, which is one of the key regions of magnetic detonation. In the case of zero volume viscosity, thermal conduction leads to an isothermal discontinuity instead of the shock, for which temperature is continuous while density and pressure experience jump.