The current understanding in the study of focused electron beam induced processing (FEBIP) is that the growth of a deposit is mainly the result of secondary electrons (SEs). This suggests that the growth rate for FEBIP is affected by the SE emission from the support. Our experiments, with membranes thinner than the SE escape depth, confirm this hypothesis. We used membranes of 1.4 and 4.3 nm amorphous carbon as supports. At the very early stage, the growth is support-dominated and the growth rate on a 4.3 nm thick membrane is three times higher than on a 1.4 nm thick membrane. This is consistent with Monte Carlo simulations for SE emission. The results suggest that SEs are dominant in the dissociation of W(CO)6 on thin membranes. The best agreement between simulations and experiment is obtained for SEs with energies between 3 and 6 eV.With this work we revisit earlier experiments, working at a precursor pressure 20 times lower than previously. Then, despite using membranes thinner than the SE escape depth, we did not see an effect on the experimental growth rate. We explain our current results by the fact that very early in the process, the growth becomes dominated by the growing deposit itself.