We investigate the contribution of inelastic and elastic processes to single inclusive hadron production in proton-proton and proton (deuteron)-nucleus collisions at RHIC and the LHC. Using the hybrid formulation which includes both elastic and inelastic contributions, supplemented with the running-coupling Balitsky-Kovchegov equation, we get a good description of RHIC data. It is shown that inclusion of the inelastic terms makes the transverse momentum dependence of the production cross section steeper in the midrapidity region but does not affect the cross section in the very forward region. The inelastic processes also lead to a sharper increase of the nuclear modification factor R-pA with increasing p(T). We also make predictions for the nuclear modification factor in proton-nucleus collisions at the LHC (root s = 4.4 and 8.8 TeV) at various rapidities using the color glass condensate framework.