The possibility of using acoustic radiation force in standing waves for stirring and mixing small volumes of liquids is theoretically analyzed. The principle of stirring considered in this paper is based on moving the microparticles suspended in a standing acoustic wave by changing the frequency so that one standing wave mode is replaced by the other, with differently positioned minima of potential energy. The period-average transient dynamics of solid microparticles and gas microbubbles is considered, and simple analytical solutions are obtained for the case of standing waves of variable amplitude. It is shown that bubbles can be moved from one equilibrium position to another two to three orders of magnitude faster than solid particles. For example, radiation force in a standing acoustic wave field may induce movement of microbubbles with a speed of the order of a few m/s at a frequency of 1 MHz and ultrasound pressure amplitude of 100 kPa, whereas the speed of rigid particles does not exceed 1 cms under the same conditions. The stirring effect can be additionally enhanced due to the fact that the bubbles that are larger and smaller than the resonant bubbles move in opposite directions. Possible applications of the analyzed stirring mechanism, such as in microarrays, are discussed.