The growth of GaSb films by molecular beam epitaxy on GaAs(001)-c(4x4) at 490 degrees C has been studied in situ with scanning tunneling microscopy and ex situ with transmission electron microscopy. As the film is deposited, four distinct growth regimes are observed : the first two monolayers grow layer by layer with platelet-like two-dimensional (2D) islands; the next monolayer forms coherently strained three-dimensional (3D) quantum dots; further deposition induces film relaxation and rough 3D growth; for film thicknesses >100 nm the growth is again 2D, proceeding via spiral growth around emerging threading dislocations. The atomic-scale mechanisms inherent in the transitions between the growth regimes are discussed. Variations in growth procedures aimed at improving the quantum dot uniformity and reducing the dislocation density are proposed.