The role of lead white is multiple. It is a bulk drier that means homogenous drying over the paint film. It reacts with free acids that develop by oxidation of the oil. It links the acids groups of C9 diacids even when they are still acylglycerides thus stabilising the early film (proven by NMR work of Michiel Verhoeven in 2006). At a later stage when the biological ester bonds are gone, it further stabilises the diacids to form a network that is hard to break. And furthermore, it provides a filling material and surfaces for oil-derived compounds to dock.
No lead means that something else has to take over these roles. I don't think that that magical material has been found yet simply because we only very recently have begun to understand the molecular structure of paints. It is my belief that the period 1950-2000 will bring us many defects in the oil paints that are now maturing. It is time that we develop a molecular level understanding on how to deal with oil paint defects that develop when the network can't be stabilised with suitable metal cations.