String perturbation theory in the presence of D-branes is usually described in terms of conformal field theory (CFT) on worldsheets with boundaries. In this formalism, closed string scattering amplitudes in the presence of static Dp-branes exhibit infrared divergences for p=0 and p=1, so the worldsheet CFT description breaks down. For p=0, it is known that these divergences are due to D0-brane recoil. A systematic framework to take recoil into account is the worldline formalism, where fixed boundary conditions are replaced by dynamical D0-brane worldlines. In this formalism, the divergences that plague the worldsheet CFT are automatically cancelled in a non-trivial way. The amplitudes derived in the worldline formalism can be reproduced by deforming the CFT with a specific bilocal recoil operator. For p=1, the divergences are due to local recoil of D1-branes, which (classically) end up displaced a finite distance from their original position. The quantum version of this phenomenon can be viewed as a simple geometric manifestation of the absence of spontaneous symmetry breaking in 1+1 dimensions. Through a Dirac-Born-Infeld analysis, it is possible to resum these divergences in a way that yields finite, momentum-conserving amplitudes.