We propose a scheme to manipulate the local orbital angular momentum (OAM) of the ultra-broadband (0.1-30 THz) terahertz (THz) waves from the laser-induced short air filament via chirping the few-cycle vortex laser pump. The simulation results show that either the THz vortex pulses with linear azimuth-dependent phases or the THz angular accelerating vortex beams (AAVBs) with nonlinear azimuth-dependent phases can be produced by tuning the chirp parameter of the pump. Thus, the dominant physical mechanism for THz generation can be determined. The THz temporal and transverse spatial distributions can be also controlled by the chirp parameter. Furthermore, their local OAM density distributions present very complex structures because most of the modulated azimuthal intensity and the corresponding local angular helicity distributions are not able to cancel out completely. Via analyzing the simulated THz results at the different pump intensities, we classify the initial pump intensity into three cases. For the low intensity case, the Kerr effect comes into prominence, so the generated THz radiation shall be vortex pulses. While for the high intensity case, the leading plasma effect dominates. In contrast, when the pump intensity is at the medium level, the Kerr nonlinearity and the plasma effect may be comparable and competitive. Basically, THz AAVBs are generated for both high and medium intensity cases. Our study will provide the possibility for studying the optically induced rotation technology more intuitively from the perspective of angular momentum transfer.