Birational geometry of some universal families of n-pointed Fano fourfolds

The object of this note are the moduli spaces of cubic fourfolds (respectively, Gushel-Mukai fourfolds) which contain some special rational surfaces. Under certain hypotheses on the families of such surfaces, we develop a general method to show the unirationality of moduli spaces of n-pointed such fourfolds. We apply this to some codimension-one loci of cubic fourfolds (respectively, Gushel-Mukai fourfolds) which appeared in the literature recently.