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Adhesive contact between a thin elastic sheet and a substrate arises in a range of biological, physical and technological applications. By considering the dynamics of this process that naturally couples fluid flow, long-wavelength elastic deformations and microscopic adhesion, we analyse a sixth-order thin-film equation for the short-time dynamics of the onset of adhesion and the long-time dynamics of a steadily propagating adhesion front. Numerical solutions corroborate scaling laws and asymptotic analyses for the characteristic waiting time for adhesive contact and for the speed of the adhesion front. A similarity analysis of the governing partial differential equation further allows us to determine the shape of a fluid-filled blister ahead of the adhesion front. Finally, our analysis reveals a near-singular behaviour at the moving elastohydrodynamic contact line with an effective boundary condition that might be useful in other related problems.