The process of tunneling injection is known to improve the dynamical characteristics of quantum well and quantum dot lasers; in the latter, it also improves the temperature performance. The advantage of the tunneling injection process stems from the fact that it avoids hot carrier injection, which is a key performance-limiting factor in all semiconductor lasers. The tunneling injection process is not fully understood microscopically and therefore it is difficult to optimize those laser structures. We present here a numerical study of the broad band carrier dynamics in a tunneling injection quantum dot gain medium in the form of an optical amplifier operating at 1.55 um. Charge carrier tunneling occurs in a hybrid state that joins the quantum dot first excited state and the confined quantum well - injection well states. The hybrid state, which is placed energetically roughly one LO phonon above the ground state and has a spectral extent of about 5 meV , dominates the carrier injection to the ground state. We calculate the dynamical response of the inversion across the entire gain spectrum following a short pulse perturbation at various wavelengths and for two bias currents. At a high bias of 200 mA, the entire spectrum exhibits gain; at 30 mA, the system exhibits a mixed gain - absorption spectrum. The carrier dynamics in the injection well is calculated simultaneously. We discuss the role of the pulse excitation wavelengths relative to the gain spectrum peak and demonstrate that the injection well responds to all perturbation wavelengths, even those which are far from the region where the tunneling injection process dominates.