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We present experimental results on proton-deuteron and deuteron-deuteron elastic scattering measured at the two highest ISR energies, √ s = 53 GeV and √ s = 63 GeV. The data cover the single- and multiple-scattering regions over a wide interval of four-momentum transfer t . In both reactions we find clear evidence for a substantial t -dependent contribution of inelastic intermediate states in the multiple-scattering region, as well as in single scattering. In the analysis we use the Glauber multiple-scattering theory extended to include inelastic shadow effects. This extension of the basic theory contains as input a triple-Regge parametrization describing the high-mass inclusive spectrum. The analysis of inelastic corrections to multiple scattering on deuterons at high energies is shown to provide a sensitive test of different parametrization of inclusive production in proton-proton collisions.