We have investigated the time scale of the excitation of electrons leading to a transition from the quantum Hall (QH) state to the dissipative state in the two-dimensional electron systems of GaAs/AlGaAs heterostructures. The measurements were performed by applying nanosecond electric pulses to the current contacts of QH devices having various electron mobilities $(1\ifmmode\times\else\texttimes\fi{}{10}^{5}<~{\ensuremath{\mu}}_{H}<~9\ifmmode\times\else\texttimes\fi{}{10}^{5}{\mathrm{cm}}^{2}/\mathrm{V}\mathrm{}\mathrm{s})$ at integer filling factors \ensuremath{\nu}=2, 4, and 6. The breakdown of the quantum Hall effect (QHE) occurs when the applied pulses exceed a critical pulse width ${t}_{p}^{c}$ that is a function of pulse amplitude, magnetic field, and electron mobility $(0.4<~{t}_{p}^{c}<~18\mathrm{ns}).$ By applying a simple drift model, we obtain an estimate for the critical drift lengths, which vary from about 1 to 25 \ensuremath{\mu}m. No general differences of the response to short electric pulses have been found between Hall bar and Corbino devices. We interpret the critical times as drifting times between inelastic scattering events, and the critical lengths as related to the inelastic scattering lengths ${\mathcal{l}}_{\mathrm{in}}$ of electrons causing the QHE breakdown. Characteristic dependences of the critical drift times and lengths on the amplitude, filling factor, and the mobility have been observed and can be attributed to impurity-assisted inter-Landau-level tunneling.