Stability issues have been a major concern for the realization of broadband tunability of the gyrotron backward-wave oscillator (gyro-BWO). Multimode, time-dependent simulations are employed to examine the stability properties of the gyro-BWO. It is shown that the gyro-BWO is susceptible to both nonstationary oscillations and axial mode competition in the course of frequency tuning. Regions of nonstationary oscillations and axial mode competition are displayed in the form of stability maps over wide-ranging parameter spaces. These maps serve as a guide for the identification and optimization of stable windows for broadband tuning. Results indicate that a shorter interaction length provides greater stability without efficiency degradation. These theoretical predictions have been verified in a Ka-band gyro-BWO experiment using both short and long interaction lengths. In the case of a short interaction length, continuous and smooth tunability, in magnetic field and in beam voltage, was demonstrated with the high interaction efficiency reported so far. A maximum 3-dB tuning range of 1.3 GHz with a peak power of 149 kW at 29.8% efficiency was achieved. In a comparative experiment with a longer interaction length, the experimental data are characterized by piecewise-stable tuning curves separated by region(s) of nonstationary oscillations, as predicted by theory.