The growth of alloy layers by molecular-beam deposition is considered theoretically, with emphasis on the connection between local variations in the composition and surface roughening. For alloy layers mismatched to the substrate, the elastic fields generated by the mismatch couple with the elastic fields generated by compositional inhomogeneities, leading to a joint compositional/morphological instability under certain growth conditions. We show that the non-equilibrium nature of the growth leads to a competition between these strain relieving mechanisms, the alloy thermodynamics and the kinetics of the deposition process. We show explicitly how our theory is applied to real systems, taking InGaAs layers as an example. The theory is generalized to include the formation of atomically ordered phases during growth, by focusing on the experimentally observed relation between ordering and local surface misorientations.