The modulation bandwidth of double tunneling-injection (DTI) quantum dot (QD) lasers is studied, taking into account noninstantaneous pumping of QDs. In this advanced type of semiconductor lasers, carriers are first captured from the bulk waveguide region into two-dimensional regions (quantum wells [QWs]); then they tunnel from the QWs into zero-dimensional regions (QDs). The two processes are noninstantaneous and, thus, could delay the delivery of the carriers to the QDs. Here, the modulation bandwidth of DTI QD lasers is calculated as a function of two characteristic times (the capture time from the waveguide region into the QW and the tunneling time from the QW into the QD ensemble) and is shown to increase as either of these times is reduced. The capture and tunneling times of 1 and 0.1 ps, respectively, are shown to characterize fast capture and tunneling processes; as the capture and tunneling times are brought below 1 and 0.1 ps, the bandwidth remains almost unchanged and close to its upper limit.