We consider the sticking of a fluid-immersed colloidal particle with a substrate coated by polymeric
tethers, a model for soft, wet adhesion in many natural and artificial systems. Our theory accounts for the
kinetics of binding, the elasticity of the tethers, and the hydrodynamics of fluid drainage between the
colloid and the substrate, characterized by three dimensionless parameters: the ratio of the viscous
drainage time to the kinetics of binding, the ratio of elastic to thermal energies, and the size of the particle
relative to the height of the polymer brush. For typical experimental parameters and discrete families of
tethers, we find that adhesion proceeds via punctuated steps, where rapid transitions to increasingly bound
states are separated by slow aging transients, consistent with recent observations. Our results also suggest
that the bound particle is susceptible to fluctuation-driven instabilities parallel to the substrate.