Long-term JHK light curves have recently become available for large numbers of the more luminous stars in the Small Magellanic Cloud (SMC). We have used these JHK light curves, along with OGLE (Optical Gravitational Lensing Experiment) V and I light curves, to examine the variability of a sample of luminous red giants in the SMC which show prominent long secondary periods (LSPs). The origin of the LSPs is currently unknown. In oxygen-rich stars, we found that while most broad-band colours (e.g. V- I) get redder when an oxygen-rich star dims during its LSP cycle, the J-K colour barely changes and sometimes becomes bluer. We interpret the J-K colour changes as being due to increasing water vapour absorption during declining light caused by the development of a layer of dense cool gas above the photosphere. This result and previous observations which indicate the development of a chromosphere between minimum to maximum light suggest that the LSP phenomenon is associated with the ejection of matter from the stellar photosphere near the beginning of light decline. We explore the possibility that broad-band light variations from the optical to the near-infrared regions can be explained by either dust absorption by ejected matter or large spots on a rotating stellar surface. However, neither model is capable of explaining the observed light variations in a variety of colour-magnitude diagrams. We conclude that some other mechanism is responsible for the light variations associated with LSPs in red giants.