1. Field of the Invention The present invention relates to a display control apparatus, a display control method, and a storage medium for displaying a moving image on a screen of a display device. 2. Description of the Related Art Conventionally, a variety of display control methods that provide an increased dynamic range and reduced color breakup have been proposed (see, for example, Japanese Patent Application Laid-Open No. 2004-289495 and Japanese Patent Application Laid-Open No. 2003-043988). Japanese Patent Application Laid-Open No. 2004-289495 describes a liquid crystal display which improves the contrast and luminance in each of a black display and a white display and thereby reduces color breakup. Japanese Patent Application Laid-Open No. 2003-043988 describes a liquid crystal display which uses multiple backlights having different spectral characteristics in order to perform gamma correction while reducing color breakup. In general, when a liquid crystal display or the like displays a moving image, its power consumption increases. On the other hand, there is a demand to increase the brightness of a screen while consuming less power. For example, Japanese Patent Application Laid-Open No. 2010-118027 proposes an image processing apparatus which uses a large-size backlight for a black display in a dark scene and uses a small-size backlight for a white display in a bright scene. Such a display can reduce power consumption by switching between the large-size backlight and the small-size backlight. Japanese Patent Application Laid-Open No. 2010-118027 also describes that when there is a scene change from a black display to a white display, the white display is performed with the large-size backlight and the black display is performed with the small-size backlight. However, although the display control method proposed in Japanese Patent Application Laid-Open No. 2010-118027 can improve the moving image display performance (color reproducibility) of the liquid crystal display by reducing power consumption, it has been found that the quality of the moving image is often inadequate. The cause of this is that the luminance of a backlight varies significantly from one display screen to another in response to a scene change. For example, when a scene changes from a black display to a white display, the liquid crystal display displays the white display with the large-size backlight and displays the black display with the small-size backlight. At this time, the power consumption of the display is reduced by only switching a backlight between the large-size and small-size backlights. Therefore, the luminance of the backlight varies greatly from one screen to another even when the backlights are identical. As a result, the contrast and brightness of a screen displayed varies from one screen to another even when there is no scene change. When there is a scene change, the quality of a moving image may deteriorate. This problem can be solved by keeping the luminance of a backlight constant by constantly switching between the large-size and small-size backlights regardless of the luminance of a display screen. However, in this case, if a scene changes to a white display, a backlight may be switched to the small-size backlight when there is a scene change from a black display to a white display, and may be switched to the large-size backlight when there is a scene change from a white display to a black display. As a result, the luminance of a backlight varies in response to a scene change even though no change in brightness occurs. Therefore, this method deteriorates the quality of the moving image more seriously than with a conventional moving image display. Furthermore, a display control apparatus may perform a variety of processing including backlight on/off switching for power consumption reduction (see, for example, Japanese Patent Application Laid-Open No. 2007-033221). However, even with such a display control apparatus, the above-described problem may not be solved.