We study the dynamics of entanglement in the scaling limit of the Ising spin chain in the presence of both a longitudinal and a transverse field. We present analytical results for the quench of the longitudinal field in critical transverse field which go beyond current lattice integrability techniques. We test these results against a numerical simulation on the corresponding lattice model finding extremely good agreement. We show that the presence of bound states in the spectrum of the field theory leads to oscillations in the entanglement entropy and suppresses its linear growth on the time scales accessible to numerical simulations. For small quenches we determine exactly these oscillatory contributions and demonstrate that their presence follows from symmetry arguments. For the quench of the transverse field at zero longitudinal field we prove that the R\'enyi entropies are exactly proportional to the logarithm of the exponential of a time-dependent function, whose leading large-time behaviour is linear, hence entanglement grows linearly. We conclude that, in the scaling limit, linear growth and oscillations in the entanglement entropies do not appear related to integrability and its breaking respectively.