Wiedemann-Franz law is not the mechanism, but, rather, an empirical fact. An empirical law doesn't explain a phenomenon on its basic mechanism/s, but just gives the relation between variables. In other words, it is the result of, not its cause.
The relationship is based upon the fact that heat and electrical transport both involve the free electrons in the metal.
Right. Both are based on the motion of (nearly) free electrons (mostly). This is of course known. My question aims at the deeper level of the mechanism itself, which make up the overall picture of transport.
The fact that the electrons are free to move IS the mechanism in this case(!)
Of course you can use more realistic (and complicated) models for the transport; but ultimately the correlation is due to the presence of free electrons; it is not more complicated than that.