Precision absolute gravity measurements are growing in importance, especially in the context of the new definition of the kilogram. For the case of free fall absolute gravimeters with a Michelson-type interferometer tracking the position of a free falling body, one of the effects that needs to be taken into account is the 'speed of light perturbation' due to the finite speed of propagation of light. This effect has been extensively discussed in the past, and there is at present a disagreement between different studies. In this work, we present the analysis of new data and confirm the result expected from the theoretical analysis applied nowadays in free-fall gravimeters. We also review the standard derivations of this effect (by using phase shift or Doppler effect arguments) and show their equivalence.