How far can free radicals from UV radiation diffuse through the skin?
The life of the radicals decides how far they move.

This paper https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4310837/#!po=6.92308 has a table of half lives of radicals, the longest lasting ones (the radical of nitrous oxide and nitrogen dioxide) have half lives of seconds.

Diffusion theough the skin would be slow

Technically the problem is "reactive oxygen species" (ROS) and "reactive nitrogen species" (RNS) that are the damaging chemicals that includes free radicals. The reactive species that are not radicals have a longer half-life or are even fairly stable and so could diffuse further before making mischief.

The wikipedia article has problems - that is mentioned on the talk page.