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It is well-known that there is a gradient, there will drive a flow inevitably. For example, a density-gradient may drive a diffusion flow, an electrical potential-gradient may drive an electric current in plasmas, etc. Then, what will be driven when a magnetic-gradient occurs in solar atmospheric plasmas? Considering the ubiquitous distribution of magnetic-gradient in solar plasma loops, this work demonstrates that magnetic-gradient pumping (MGP) mechanism is valid even in the partial ionized solar photosphere, chromosphere as well as in the corona. It drives energetic particle flows which carry and convey kinetic energy from the underlying atmosphere to move upwards, accumulate around the looptop and increase there temperature and pressure, and finally lead to eruptions around the looptop by triggering ballooning instabilities. This mechanism may explain the formation of the observing hot cusp-structures above flaring loops in most preflare phases, therefore, the magnetic-gradient should be a natural driver of solar eruptions. Furthermore, we may also apply to understand many other astrophysical phenomena, such as the temperature distribution above sunspots, the formation of solar plasma jets, type-II spicule, and fast solar wind above coronal holes, as well as the fast plasma jets related to white dwarfs, neutron stars and black holes.