We report on the use of time resolved X-ray diffraction to measure the dynamics of strain in laser-excited graphite film of nanometer thickness, obtained by chemical vapour deposition (CVD). Heat transport in the CVD film is simulated with a 1-dimensional heat diffusion model. We find the experimental data to be consistent with a c-axis thermal conductivity of ∼0.7 W m⁻¹ K⁻¹. This value is four orders of magnitude lower than the thermal conductivity in-plane, confirming recent theoretical calculations of the thermal conductivity of multilayer graphene.