High performance of both photovoltaic and electroluminescent devices requires low nonradiative recombination losses. In perovskites, such loses strongly depend on the carrier traps related to the mobile ions and vacancies, causing I-V hysteresis of solar cells and influencing the performance of other optoelectronic devices, such as photodetectors and LEDs. To address the dynamics of the mobile ions, here we investigate electroluminescence time evolution in perovskite solar cells under constant and pulsed voltage conditions. We propose a model, accounting for the spatial ion accumulation and explaining the complex electroluminescence dynamics both on fast (microseconds) and slow (seconds) time scales. We demonstrate the appearance of a high-intensity short electroluminescence peak (overshoot pulse) immediately after termination of the electrical pulse. The generation of a giant overshoot pulse suggests a simple way to achieve high pulsed luminescence intensity with a low current density, which opens new prospects toward optical gain and implementation of electrically pumped lasers.