We investigated by tight-binding molecular dynamics the structure, the bulk modulus and the electronic properties of the smallest four-branched carbon schwarzites fcc-(C28)2, fcc-(C36)2 and fcc-(C40)2 having the shape of a D minimal periodic surface. They are found to have a stability comparable to that of fullerene C60 and to exhibit alternative metallic and insulating characters, with an apparent relationship to their local geometry. We also studied the coalescence of fullerenic fragments and carbon clusters by following the evolution of the topological connectivity. Though different temperature variation protocols lead to irregular structures similar to random schwarzites, their connectivity is found to stabilize at values corresponding either to tubulene or three-branched schwarzites, indicating that long time evolution at constant connectivity is potentially able to yield regular shapes. Experiments on laser-induced transformations of fullerite occasionally yield branched tubular structures with a schwarzite shape.