We report on the reaction of surface-attached polymer brushes with polymer molecules, which are present in a contacting solution. More specifically, we synthesize copolymer brushes containing N-hydroxysuccinimide active ester groups and study their reaction with poly(ethylene glycol) molecules, which carry an amino group at one end. Qualitatively, highly "pegylated" surfaces are obtained with an equivalent thickness of the PEG layers of up to 200 nm. Yield and kinetics of the grafting reaction are studied as a function of size and concentration of the incoming molecules, brush graft density, and brush molecular weight. It is observed that the completion of the grafting reaction is surprisingly fast, the molecular weight of the incoming molecules has a strong, while graft density and molecular weight of the brush have only a weak influence on the extent of functionalization. We develop a model for the grafting reaction, where due to limited penetration of the brush by the incoming chains the reaction occurs only at the outer periphery of the surface-attached layer. Reactive groups present in the inner parts of the brush do not take part in the attachment reaction. In the outer parts of the brush, however, partial penetration is possible so that the attachment reaction is fast and essentially only governed by the details of the shape of the segment density distribution and the size of the incoming chains.