Epitaxial growth of three-dimensional SiGe islands on Si(100) has been investigated by low-energy electron microscopy. Contrast mechanisms that allow measuring the evolution in position, size and shape of individual islands during deposition have been identified, and were used to study the epitaxial embedding of faceted SiGe islands in a Si matrix. Deposition of Si onto SiGe islands drives their expansion and causes a change in their shape: the island apex decays and a (100) top facet forms. A physical interpretation of the mechanism producing this shape change is presented. The eventual smoothing of the growth front upon embedding of the islands is promoted by the strain-driven migration of adatoms away from the (100) top facet.